Elegant anti-disturbance control for stochastic systems with multiple heterogeneous disturbances based on fuzzy logic systems

Abstract

There are a large number of non-harmonic disturbances generated by nonlinear exogenous systems in realistic engineering. The current disturbance observer is not applicable for estimating the non-harmonic disturbance with unknown nonlinear dynamics, thus greatly reducing the accuracy of the controller. This paper addresses a class of stochastic systems with multiple heterogeneous disturbances including white noise and non-harmonic disturbance with unknown smooth nonlinear function, which can be approximated by fuzzy logic systems. Based on the approximation of the unknown nonlinear function, an adaptive disturbance observer (ADO) is constructed to estimate non-harmonic disturbance. Combining disturbance observer-based control with fuzzy control, an elegant anti-disturbance control (EADC) scheme is proposed such that the composite system achieves asymptotically bounded in mean square. Simulation examples show that the state responses of the system gradually approache $\pm 0.05$ from divergence, indicating that the effectiveness of the controller is satisfactory. In addition, the anti-disturbance control accuracy of EADC approximately improves $70 ~ 200$ times compared with $H_{\infty}$ control. The simulation results demonstrate the feasibility and effectiveness of the proposed scheme.

Keywords

Adaptive disturbance observer elegant anti-disturbance control multiple heterogeneous disturbances fuzzy logic systems stochastic system

Introduction

In a complex environment, disturbances of system include measured noise, external environment disturbance, change of the system structure and modeling error (Han et al., 2018; Hu et al., 2019; Jiao et al., 2019; Wei and Sun, 2018; Zhang et al., 2015). Since various types of disturbances always effect the control accuracy of the system, the problem of anti-disturbance control for systems with multiple disturbances has been a hot topic in the control field (Liu et al., 2018; Sun et al., 2018; Wei et al., 2019, 2016). Most anti-disturbance control methods focus on integrating multiple disturbances into an equivalent system disturbance, which do not make full use of the influence mechanism and the characteristics of different disturbances (Guo and Chen, 2005; Jiao et al., 2016a, 2016b; Yang et al., 2013). In recent years, Guo and Cao (2014) presented a composite hierarchical anti-disturbance control (CHADC) structure, which can reject the inner loop disturbances and attenuate the outer loop ones separately and simultaneously. In the work of Wei and Chen (2014), a composite hierarchical anti-disturbance control scheme was presented for a class of nonlinear systems with unknown nonlinear dynamics and disturbances, which include partially-known information and the bounded $H_{2}$ norm. A class of discrete-time stochastic systems with the disturbances were considered, which consist of the disturbance with partially-known information and white noise (Wei et al., 2018). Considering different characteristics of disturbances with different types of modelling, elegant anti-disturbance control (EADC) means to take full advantage of the information of disturbances to obtain better the anti-disturbance performances, the above-mentioned CHADC is exactly a special case of EADC.

In the above investigations, the disturbances with partially known information are supposed to be harmonic, constant or neutral stable, which can be described by linear exogenous system. In fact, there are a number of non-harmonic disturbances, which cannot be described by linear exogenous system (Ding, 2006; Lu and Huang, 2015). Generally speaking, the nonlinear function in non-harmonic disturbances is considered to be known (Dong et al., 2019; Wang et al., 2016). In Dong et al. (2019), a stochastic nonlinear disturbance observer was constructed to estimate the non-harmonic disturbances with known nonlinear functions. However, the accurate nonlinearity information of non-harmonic disturbances is not easy to get in many practical engineering applications. Hence, it is necessary to consider the non-harmonic disturbances with unknown nonlinear functions. Note that the current method proposed by Dong et al. (2019) is not applicable for estimating the non-harmonic disturbances with unknow nonlinear function, thus greatly reducing the accuracy of the controller. Since fuzzy logic systems do not need the accurate mathematical model of the system, which provides a possibility to deal with above-mentioned challenge (Chen et al., 2014; Su and Zhang, 2019; Wei and Chen, 2014).

In recent years, fuzzy logic systems have been successfully used to approximate unknown nonlinear functions for stochastic systems (Sui et al., 2015; Wang et al., 2019; Zhao et al., 2018). In Sui et al. (2015), a class of nonlinear stochastic systems including unknown nonlinear uncertainties, the input saturation, unmodeled dynamics and unmeasured states were considered. A command-filter-based adaptive fuzzy control method was presented to solve stochastic disturbances and input saturation problems (Zhao et al., 2018). Since different sources and channels of the disturbances, the above methods cannot be directly extended to deal with multiple heterogeneous disturbances.

Based on the above discussions, the purpose of this paper is to propose an EADC scheme for a class of stochastic systems with multiple heterogeneous disturbances. The main contributions of this paper are summarized as follows:

(1) The non-harmonic disturbances modeled by nonlinear exogenous systems with unknown smooth nonlinear functions are considered in this paper. Fuzzy logic systems are utilized to approximate the unknown smooth nonlinear functions. Compared with Dong et al. (2019), the proposed method has a wider application range.

(2) An adaptive disturbance observer (ADO) is constructed to estimate the non-harmonic disturbances based on the approximation of the unknown nonlinear function. Combining disturbance observer-based control with fuzzy control, an EADC scheme that can make full use of disturbance information is proposed to achieve higher anti-disturbance control accuracy.

The main content of this paper includes the following parts. In Section 2, the formulation of the problem is given. The ADO and the EADC scheme are designed in Section 3. In Section 4, numerical simulation and application example are given. Section 5 is the summary of this paper.

Formulation of the problem

The stochastic system with multiple heterogeneous disturbances is represented as

\overset{\cdot}{x} (t) = Ax (t) + B_{0} [u (t) + D_{0} (t)] + B_{1} ξ_{1} (t) + B_{2} x (t) ξ_{2} (t),

(1)

where $x (t) \in R^{n}$ and $u (t) \in R^{m}$ are the system state and the control input, respectively. The additive noise $ξ_{1} (t) \in R$ and multiplicative noise $ξ_{2} (t) \in R$ are band-limited white noises. $A \in R^{n \times n}$ , $B_{0} \in R^{n \times m}$ , $B_{1} \in R^{n \times 1}$ and $B_{2} \in R^{n \times n}$ are the parameter matrices. $D_{0} (t) \in R^{m}$ represents the non-harmonic disturbance with unknown smooth nonlinear function, which can be formulated by following nonlinear exogenous systems

{\begin{matrix} \overset{\cdot}{z} (t) = Wz (t) + Ff (z (t)), \\ D_{0} (t) = Vz (t), \end{matrix}

(2)

where $z (t) \in R^{2 r}$ represents the state vector of the exogenous systems. $W$ , $F$ and $V$ are the known coefficient matrices. $f (z (t)) \in R$ is an unknown smooth nonlinear function, which can be approximated by fuzzy logic systems. And the fuzzy logic systems satisfy the following if-then rules (Chen et al., 2014)

$R^{l}$ : if $z_{1}$ is $H_{1}^{l}, \dots$ and if $z_{n}$ is $H_{n}^{l}$ , then $y$ is $G^{l}$ , $l = 1, 2, \dots, N$

where $z = [z_{1}, \dots, z_{n}]$ and $y$ are the fuzzy logic systems input and output, respectively. $H_{i}^{l} (i = 1, 2, \dots, n)$ and $G^{l}$ are fuzzy sets associated with the membership functions $μ_{H_{i}^{l}} (z_{i})$ and $μ_{G^{l}} (y)$ , respectively. $N$ is the number of the rules.

The fuzzy logic systems with the singleton fuzzifier, product inference and center average defuzzifier are presented as

y (z) = \frac{\sum_{l = 1}^{N} {\bar{y}}_{l} Π_{i = 1}^{n} μ_{H_{i}^{l}} (z_{i})}{\sum_{l = 1}^{N} [Π_{i = 1}^{n} μ_{H_{i}^{l}} (z_{i})]},

where ${\bar{y}}_{l} = ma x_{y \in R} μ_{G^{l}} (y)$ . Define the fuzzy basis function as

φ_{l} (z) = \frac{Π_{i = 1}^{n} μ_{H_{i}^{l}} (z_{i})}{\sum_{l = 1}^{N} [Π_{i = 1}^{n} μ_{H_{i}^{l}} (z_{i})]} .

Set $θ = [{\bar{y}}_{1}, {\bar{y}}_{2}, \dots, {\bar{y}}_{N}]^{T} = [θ_{1}, θ_{2}, \dots, θ_{N}]^{T}$ , and $φ (z) = [φ_{1} (z), φ_{2} (z), \dots, φ_{N} (z)]^{T}$ . Then, the fuzzy logic systems can be formulated as follows

y (z) = θ^{T} φ (z) .

(3)

On the basis of Lee and Tomizuka (2000), the aforementioned fuzzy logic systems $y (z) = \hat{f} (z | θ)$ can be used to approximate the continuous function $f (z)$ in following form

\hat{f} (z | θ) = θ^{T} φ (z) .

(4)

Define the optimal parameter vector $θ^{*}$ as

θ^{*} = \underset{θ \in Ω_{1}}{\arg min} [sup_{z \in Ω_{z}} ‖ \hat{f} (z | θ) - f (z | θ) ‖],

(5)

where $Ω_{1}$ and $Ω_{z}$ are compact regions for $θ$ and $z$ , respectively. The fuzzy minimum approximation error is defined as

ε (z) = f (z) - \hat{f} (z | θ^{*}) .

(6)

On the basis of (3)–(6), states of the non-harmonic disturbances can be rewritten as

\begin{matrix} \overset{\cdot}{z} (t) = Wz (t) + F [\hat{f} (z | θ) + (\hat{f} (z | θ^{*}) - \hat{f} (z | θ)) + (f (z) - \hat{f} (z | θ^{*}))] \\ = Wz (t) + F θ^{T} φ (z) + F {\tilde{θ}}^{T} φ (z) + F ε (z), \end{matrix}

(7)

where $\tilde{θ} = θ^{*} - θ$ , disturbance model (2) can be described as

{\begin{matrix} \overset{\cdot}{z} (t) = Wz (t) + F θ^{T} φ (z) + F {\tilde{θ}}^{T} φ (z) + F ε (z), \\ D_{0} (t) = Vz (t) . \end{matrix}

(8)

Remark 1: In fact, the disturbance model in (2) can represent a large type of disturbances. When $F = 0$ , $D_{0} (t)$ can represent unknown constant disturbance and harmonics disturbance with unknown phase and amplitude (Chen and Chen, 2010; Guo and Chen, 2005). For the case with $F \neq 0$ , in most existing results, $D_{0} (t)$ can represent non-harmonic disturbances generated by nonlinear exogenous systems (Dong et al., 2019; Lu and Huang, 2015; Wang et al., 2016). However, in these results, the nonlinear function in non-harmonic disturbances is known. However, it is difficult to get accurate nonlinear information of non-harmonic disturbances in practice. Hence, the nonlinear function $f (z (t))$ is considered to be unknown in (2).

Remark 2: The block diagram of EADC is shown in Figure 1.

The disturbances in system (1) are divided into the disturbances with partially-known information and other disturbances. Because of the existence of the unknown nonlinear function $f (z (t))$ in non-harmonic disturbance $D_{0} (t)$ , it is necessary to use fuzzy logic systems to approximate it. Based on the approximation of the unknown nonlinear function, the ADO is constructed to estimate the disturbances with partially-known information online, and then compensate disturbances by combining a feedforward compensator with control law $Kx (t)$ . The feedback controller is used to attenuate the other disturbances. This structure of controller improves the control accuracy greatly, and the controller has the advantages of simple structure, easy realization and higher control accuracy.

According to Øksendal (2003), by replacing $ξ_{1} (t)$ with $\frac{d W_{1} (t)}{dt}$ and $ξ_{2} (t)$ with $\frac{d W_{2} (t)}{dt}$ , (1) and (8) have

{\begin{matrix} dx (t) = Ax (t) dt + B_{0} D_{0} (t) dt + B_{0} u (t) dt + B_{1} d W_{1} (t) + B_{2} x (t) d W_{2} (t), \\ dz (t) = Wz (t) dt + F θ^{T} φ (z) dt + F {\tilde{θ}}^{T} φ (z) dt + F ε (z) dt, \\ D_{0} (t) = Vz (t), \end{matrix}

(9)

where $W_{1} (t)$ and $W_{2} (t)$ are independent standard Wiener processes (Brownian motion) (Hu, 2017; Hu et al., 2014) on $(Ω, F, F_{t}, P)$ with a filtration $F_{t}$ satisfying the general conditions.

Assumption 1: $(W, B_{0} V)$ is observable and $(A, B_{0})$ is controllable.

As a special case of Mao and Yuan (2006), the criterion of the stability for stochastic system is given. Consider a nonlinear stochastic differential equation (SDE)

dx (t) = f (x (t), t) dt + g (x (t), t) dB (t), t ⩾ t_{0} (⩾ 0),

(10)

where $f : R^{n} \times R_{+} \to R^{n}$ and $g : R^{n} \times R_{+} \to R^{n \times m}$ are locally Lipschitz in $x (t) \in R^{n}$ with $f (0, t) = 0$ , $g (0, t) = 0$ . $B (t)$ , $t ⩾ 0$ is an m-dimensional independent standard Wiener process.

Lemma 1: (Mao and Yuan, 2006). Assume there exists functions $V \in C^{2.1} (R^{n} \times R_{+})$ and $κ \in κ_{ν} \subset κ_{\infty}$ and positive numbers $p, β, λ$ such that

κ (| x |^{p}) ⩽ V (x, t), and LV (x, t) ⩽ - λ V (x, t) + β,

(11)

for all $(x, t) \in R^{n} \times R_{+}$ . Then

E {V (\bar{x} (t), t)} ⩽ V ({\bar{x}}_{0}) e^{- μ t} + \frac{β}{α},

(12)

lim_{t \to \infty} supE | x (t; t_{0}, x_{0}) |^{p} ⩽ κ^{- 1} (\frac{β}{λ}),

(13)

for all $(x, t) \in R^{n} \times R_{+} .$ That is, system (10) is asymptotically bounded in $pth$ moment.

Main results

In this section, a novel ADO is constructed to estimate the non-harmonic disturbances based on the approximation of the unknown nonlinear function. Then, an EADC is designed.

ADO

The ADO is structured as

{\begin{matrix} dv (t) = (W + L B_{0} V) \hat{z} (t) dt + F θ^{T} \hat{φ} (z) dt + L [Ax (t) + B_{0} u (t)] dt, \\ \hat{z} (t) = v (t) - Lx (t), \\ {\hat{D}}_{0} (t) = V \hat{z} (t), \end{matrix}

(14)

where ${\hat{D}}_{0} (t)$ is the estimation of $D_{0} (t)$ , $L$ is the observer gain to be given, $v (t)$ is the state of the ADO and the fuzzy basis function $\hat{φ} (z) = φ \hat{(} z)$ .

The disturbance estimation error is denoted as $e_{z} (t) = z (t) - \hat{z} (t)$ . Based on (9) and (14), it is shown that the error dynamics satisfies

\begin{matrix} d e_{z} (t) = (W + L B_{0} V) e_{z} (t) dt + F {\tilde{θ}}^{T} \hat{φ} (z) dt + L B_{2} x (t) d W_{2} (t) \\ + F θ^{T} φ (z) dt - F θ^{T} \hat{φ} (z) dt + F ε (z) dt + L B_{1} d W_{1} (t) . \end{matrix}

(15)

Let $\bar{ε} (z) = θ^{T} \tilde{φ} (z) + ε (z)$ , where $\tilde{φ} (z) = φ (z) - \hat{φ} (z)$ . Then, we have

\begin{matrix} d e_{z} (t) = (W + L B_{0} V) e_{z} (t) dt + F {\tilde{θ}}^{T} \hat{φ} (z) dt + F \bar{ε} (z) dt \\ + L B_{1} d W_{1} (t) + L B_{2} x (t) d W_{2} (t) . \end{matrix}

(16)

According to Assumption 1, since $(W, B_{0} V)$ is observable, the pole of $d e_{z} (t) = (W + L B_{0} V) e_{z} (t) dt$ in (15) can be placed at the left-hand side complex plane (Willems and Willems, 1976; Zhang and Chen, 2004). By using the MATLAB command for pole placement $L = place (W^{T}, - V^{T} B_{0}^{T}, J)$ , where $J$ is the matrix consisting of the desired poles, we can adjust $L$ to satisfy the performance requirement for ADO.

In the following, the EADC based on the ADO is introduced as

u (t) = - {\hat{D}}_{0} (t) + Kx (t),

(17)

where $K$ is the controller gain can be solved by linear matrix inequality (LMI). Substituting (17) into (9), the closed-loop system is given by

\begin{matrix} dx (t) = Ax (t) dt + B_{0} Vz (t) dt + B_{0} Kx (t) dt \\ + B_{1} d W_{1} (t) + B_{2} x (t) d W_{2} (t) . \end{matrix}

(18)

Combining (18) with (16), yields

\begin{matrix} [\begin{matrix} dx (t) \\ d e_{z} (t) \end{matrix}] = [\begin{matrix} A + B_{0} K & B_{0} V \\ 0 & W + L B_{0} V \end{matrix}] [\begin{matrix} x (t) \\ e_{z} (t) \end{matrix}] dt + [\begin{matrix} B_{1} \\ L B_{1} \end{matrix}] d W_{1} (t) \\ + [\begin{matrix} 0 \\ F \end{matrix}] {\tilde{θ}}^{T} \hat{φ} (z) dt + [\begin{matrix} 0 \\ F \end{matrix}] \bar{ε} (z) dt + [\begin{matrix} B_{2} & 0 \\ L B_{2} & 0 \end{matrix}] x (t) d W_{2} (t) . \end{matrix}

(19)

The composite system can be described by

\begin{matrix} d (\bar{x} (t)) = \bar{A} \bar{x} (t) dt + {\bar{B}}_{1} d W_{1} (t) + {\bar{B}}_{2} \bar{x} (t) d W_{2} (t) \\ + {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) dt + {\bar{F}}_{2} \bar{ε} (z) dt, \end{matrix}

(20)

where

\bar{x} (t) = [\begin{matrix} x (t) \\ e_{z} (t) \end{matrix}], \bar{A} = [\begin{matrix} A + B_{0} K & B_{0} V \\ 0 & W + L B_{0} V \end{matrix}], {\bar{B}}_{1} = [\begin{matrix} B_{1} \\ L B_{1} \end{matrix}],

{\bar{B}}_{2} = [\begin{matrix} B_{2} & 0 \\ L B_{2} & 0 \end{matrix}], {\bar{F}}_{1} = [\begin{matrix} 0 \\ F \end{matrix}], {\bar{F}}_{2} = [\begin{matrix} 0 \\ F \end{matrix}] .

(21)

EADC

In this section, the main purpose is to propose an EADC scheme to achieve the expected dynamic performance of the composite system. The following result can be obtained.

Theorem 1: Considering stochastic system (1) with disturbance (2) under Assumption 1, for given parameters $α > 0$ , if there exist symmetric matrices $Q_{1} = P_{1}^{- 1} > 0$ , $Q_{2} = P_{2}^{- 1} > 0$ and matrix $R$ satisfying the following LMI

Ω = [\begin{matrix} Θ_{1} & Q_{1} B_{2}^{T} & Q_{1} B_{2}^{T} L^{T} & 0 & B_{0} V Q_{2} \\ * & - Q_{1} & 0 & 0 & 0 \\ * & * & - Q_{2} & 0 & 0 \\ * & * & * & - α I & F^{T} \\ * & * & * & * & Θ_{2} \end{matrix}] < 0,

(22)

where

Θ_{1} = A Q_{1} + Q_{1}^{T} A^{T} + B_{0} R + R^{T} B_{0}^{T},

Θ_{2} = Q_{2} W + W^{T} Q_{2}^{T} + L B_{0} V Q_{2} + Q_{2}^{T} V^{T} B_{0}^{T} L^{T},

then, by designing ADO (14) with gain $L$ , EADC law (17) with gain $K = {RQ}_{1}^{- 1}$ , and the following adaptive law

\overset{\cdot}{θ} = Γ^{- 1} \hat{φ} (z) \tilde{x} (t)^{T} P {\bar{F}}_{1},

(23)

where symmetric matrix $Γ > 0$ , $\tilde{x} (t)^{T} = [x (t) \hat{z} (t)]$ , the composite system (20) is asymptotically bounded in mean square.

Proof: For the composite system (20), choose the following Lyapunov function

V (\bar{x} (t), t) = {\bar{x}}^{T} (t) P \bar{x} (t) + {\tilde{θ}}^{T} Γ \tilde{θ} .

(24)

Select

P = [\begin{matrix} P_{1} & 0 \\ 0 & P_{2} \end{matrix}] = [\begin{matrix} Q_{1}^{- 1} & 0 \\ 0 & Q_{2}^{- 1} \end{matrix}] > 0 .

(25)

Based on (24) and (25), along with the trajectories of (20), it can be shown that

\begin{matrix} LV (\bar{x} (t), t) = \frac{\partial V}{\partial \bar{x}} (\bar{A} \bar{x} (t) + {\bar{F}}_{1} {\tilde{θ}}^{T} φ (z) + {\bar{F}}_{2} \bar{ε} (z)) + Tr ({\bar{B}}_{1}^{T} P {\bar{B}}_{1}) \\ + Tr ({\bar{x}}^{T} (t) {\bar{B}}_{2}^{T} P {\bar{B}}_{2} \bar{x} (t)) + 2 {\tilde{θ}}^{T} Γ (\overset{\cdot}{θ} - \overset{\overset{\cdot}{^}}{θ}) \\ = {\bar{x}}^{T} (t) (P \bar{A} + {\bar{A}}^{T} P) \bar{x} (t) + 2 {\bar{x}}^{T} (t) P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) + 2 {\bar{x}}^{T} (t) P {\bar{F}}_{2} \bar{ε} (z) \\ + Tr ({\bar{x}}^{T} (t) {\bar{B}}_{2}^{T} P {\bar{B}}_{2} \bar{x} (t)) + Tr ({\bar{B}}_{1}^{T} P {\bar{B}}_{1}) + 2 {\tilde{θ}}^{T} Γ (0 - \overset{\overset{\cdot}{^}}{θ}) \\ ⩽ {\bar{x}}^{T} (t) (P \bar{A} + {\bar{A}}^{T} P + {\bar{B}}_{2}^{T} P {\bar{B}}_{2}) \bar{x} (t) + Tr (B_{1}^{T} P B_{1}) - 2 {\tilde{θ}}^{T} Γ \overset{\cdot}{θ} \\ + 2 {\bar{x}}^{T} (t) P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) + 2 {\bar{x}}^{T} (t) P {\bar{F}}_{2} \bar{ε} (z) . \\ ⩽ {\bar{x}}^{T} (t) (P \bar{A} + {\bar{A}}^{T} P + {\bar{B}}_{2}^{T} P {\bar{B}}_{2} + α^{- 1} P {\bar{F}}_{2} {\bar{F}}_{2}^{T} P) \bar{x} (t) + α \bar{ε} (z) {\bar{ε}}^{T} (z) \\ + 2 {\bar{x}}^{T} (t) P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) + Tr ({\bar{B}}_{1}^{T} P {\bar{B}}_{1}) - 2 {\tilde{θ}}^{T} Γ \overset{\cdot}{θ} \\ = {\bar{x}}^{T} (t) (P \bar{A} + {\bar{A}}^{T} P + {\bar{B}}_{2}^{T} P {\bar{B}}_{2} + α^{- 1} P {\bar{F}}_{2} {\bar{F}}_{2}^{T} P) \bar{x} (t) + α \bar{ε} (z) {\bar{ε}}^{T} (z) \\ + 2 (I_{1} \bar{x} (t) + I_{2} \bar{x} (t))^{T} P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) + Tr ({\bar{B}}_{1}^{T} P {\bar{B}}_{1}) - 2 {\tilde{θ}}^{T} Γ \overset{\cdot}{θ} \\ = {\bar{x}}^{T} (t) (P \bar{A} + {\bar{A}}^{T} P + {\bar{B}}_{2}^{T} P {\bar{B}}_{2} + α^{- 1} P {\bar{F}}_{2} {\bar{F}}_{2}^{T} P) \bar{x} (t) + α \bar{ε} (z) {\bar{ε}}^{T} (z) \\ + 2 [x (t) \hat{z} (t)]^{T} P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) + 2 [0 z (t)]^{T} P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) \\ + Tr ({\bar{B}}_{1}^{T} P {\bar{B}}_{1}) - 2 {\tilde{θ}}^{T} Γ \overset{\cdot}{θ} \\ = {\bar{x}}^{T} (t) (P \bar{A} + {\bar{A}}^{T} P + {\bar{B}}_{2}^{T} P {\bar{B}}_{2} + α^{- 1} P {\bar{F}}_{2} {\bar{F}}_{2}^{T} P) \bar{x} (t) + α \bar{ε} (z) {\bar{ε}}^{T} (z) \\ + 2 {\tilde{x}}^{T} (t) P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) + 2 [0 z (t)]^{T} P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) \\ + Tr ({\bar{B}}_{1}^{T} P {\bar{B}}_{1}) - 2 {\tilde{θ}}^{T} Γ \overset{\cdot}{θ}, \end{matrix}

(26)

where

I_{1} = [\begin{matrix} I & 0 \\ 0 & 0 \end{matrix}], I_{2} = [\begin{matrix} 0 & 0 \\ 0 & I \end{matrix}], \tilde{x} (t) = [\begin{matrix} x (t) \\ \hat{z} (t) \end{matrix}] .

(27)

Let

\begin{matrix} Ω_{1} = P \bar{A} + {\bar{A}}^{T} P + {\bar{B}}_{2}^{T} P {\bar{B}}_{2} + α^{- 1} P {\bar{F}}_{2} {\bar{F}}_{2}^{T} P, \\ γ (t) = α \bar{ε} (z) {\bar{ε}}^{T} (z) + Tr ({\bar{B}}_{1}^{T} P {\bar{B}}_{1}) + 2 [0 z (t)]^{T} P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) . \end{matrix}

(28)

Define

{\tilde{x}}^{T} (t) P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z) = {\tilde{θ}}^{T} Γ \overset{\cdot}{θ} .

(29)

Then, the adaptive law is designed as

\overset{\cdot}{θ} = Γ^{- 1} \hat{φ} (z) {\tilde{x}}^{T} (t) P {\bar{F}}_{1} .

(30)

From (28), there exists a constant $β ⩾ 0$ , such that $0 ⩽ γ (t) ⩽ β$ , since that $\bar{ε} (z)$ is a bounded error, $B_{1}^{T}$ and $P$ are bounded matrices, also considering that $z (t)$ , ${\bar{F}}_{1}$ , ${\tilde{θ}}^{T}$ and $\hat{φ} (z)$ are bounded matrices. Hence, we have

LV (\bar{x} (t), t) ⩽ {\bar{x}}^{T} (t) Ω_{1} \bar{x} (t) + γ (t) ⩽ {\bar{x}}^{T} (t) Ω_{1} \bar{x} (t) + β .

(31)

In the next step, we will proof $Ω < 0 \Leftrightarrow Ω_{1} < 0$ .

(1) To prove $Ω_{1} < 0 \Leftrightarrow Ω_{2} < 0$ . Based (21), (28) and schur complement formula, $Ω_{1} < 0$ is equivalent to $Ω_{2} < 0$ , where

Ω_{2} = [\begin{matrix} Π_{11} & P_{1} B_{0} V & B_{2}^{T} & B_{2}^{T} L^{T} & 0 \\ * & Π_{22} & 0 & 0 & P_{2} F \\ * & * & - P_{1}^{- 1} & 0 & 0 \\ * & * & * & - P_{2}^{- 1} & 0 \\ * & * & * & * & - α I \end{matrix}] < 0,

(32)

with

Π_{11} = P_{1} A + A^{T} P_{1} + P_{1} B_{0} K + K^{T} B_{0}^{T} P_{1},

Π_{22} = P_{2} W + W^{T} P_{2} + P_{2} L B_{0} V + V^{T} B_{0}^{T} L^{T} P_{2} .

(2) To prove $Ω_{2} < 0 \Leftrightarrow Ω_{3} < 0$ . $Ω_{2} < 0$ is simultaneously pre-multiplied and post-multiplied by diag ${Q_{1}, Q_{2}, I, I, I}$ , it is equivalent to $Ω_{3} < 0$ , where

Ω_{3} = [\begin{matrix} Θ_{1} & Q_{1} B_{2}^{T} & Q_{1} B_{2}^{T} L^{T} & 0 & B_{0} V Q_{2} \\ * & - Q_{1} & 0 & 0 & 0 \\ * & * & - Q_{2} & 0 & 0 \\ * & * & * & - α I & F^{T} \\ * & * & * & * & Θ_{2} \end{matrix}] < 0,

(33)

where

Θ_{1} = A Q_{1} + Q_{1} A^{T} + B_{0} K Q_{1} + Q_{1} K^{T} B_{0}^{T},

Θ_{2} = W Q_{2} + Q_{2}^{T} W^{T} + L B_{0} V Q_{2} + Q_{2}^{T} V^{T} B_{0}^{T} L^{T} .

(3) To prove $Ω_{3} < 0 \Leftrightarrow Ω < 0$ . It can be seen that $Ω_{3} < 0 \Leftrightarrow Ω < 0$ based on $K = {RQ}_{1}^{- 1}$ in (33).

From (1)-(3), it can be seen that $Ω < 0 \Leftrightarrow Ω_{3} < 0 \Leftrightarrow Ω_{2} < 0 \Leftrightarrow Ω_{1} < 0$ . Hence, If $Ω_{1} < 0$ holds, there exists a constant $α > 0$ such that

Ω_{1} < 0 \Rightarrow Ω_{1} + α I < 0 .

(34)

Based on (24), (25), (31) and (34), choose $κ_{\infty}$ functions $α_{1} (∥ \bar{x} (t) ∥) = λ_{\min} (P) ∥ \bar{x} (t) ∥^{p}$ and $α_{2} (∥ \bar{x} (t) ∥) = λ_{\max} (P) ∥ \bar{x} (t) ∥^{p}$ and positive number $p = 2$ , such that for all $\bar{x} (t) \in R^{n}, t ⩾ 0$

λ_{\min} (P) ∥ \bar{x} (t) ∥^{2} ⩽ V (\bar{x} (t), t) = {\bar{x}}^{T} (t) P \bar{x} (t) ⩽ λ_{\max} (P) ∥ \bar{x} (t) ∥^{2},

(35)

and choose class $κ$ functions $α_{3} (∥ \bar{x} (t) ∥) = α ∥ \bar{x} (t) ∥^{p}$ and $μ = \frac{α}{λ_{\max} (P)}$ , positive number $p = 2$ , such that for all $\bar{x} (t) \in R^{n}, t ⩾ 0$

LV (\bar{x} (t), t) ⩽ - μ V (\bar{x} (t), t) + β .

(36)

Based on Lemma 1, we have

E {V (\bar{x} (t), t)} ⩽ V ({\bar{x}}_{0}) e^{- μ t} + \frac{β}{α},

(37)

lim_{t \to \infty} supE | \bar{x} (t; t_{0}, {\bar{x}}_{0}) | ⩽ \frac{β}{μ λ_{max} (P)} = \frac{β}{α} .

(38)

Hence, by selecting EADC law (17), the adaptive law (23) and ADO (14), the composite system (20) is asymptotically bounded in mean square. The proof is completed.

Simulation examples

Numerical simulation

The numerical simulation example is given to demonstrate the efficiency of the proposed EADC scheme. Consider the stochastic system (1) with parameters as follows

\begin{matrix} A = [\begin{matrix} 1 & 32.37 & 32.2 \\ - 0.2 & 0.5 & 0 \\ 3 & 4.72 & 0 \end{matrix}], \\ B_{0} = [\begin{matrix} 3 \\ 1 \\ 2 \end{matrix}], B_{2} = [\begin{matrix} - 0.1 & - 2.65 & 0.1 \\ 0 & - 1 & 0.62 \\ 2.13 & - 0.65 & 2 \end{matrix}] . \end{matrix}

The non-harmonic disturbance $D_{0} (t)$ is formulated by (2) with

W = [\begin{matrix} - 1 & 5 \\ - 5 & - 4 \end{matrix}], V = [\begin{matrix} 5 & 0 \end{matrix}], F = [\begin{matrix} - 1.72 \\ 0.59 \end{matrix}] .

Disturbance model (2) may be affected by nonlinearity and uncertainly, similar to (Guo and Chen, 2005), it is supposed that

f (z) = 2 * \sin (z_{1}) * z_{2}^{2} .

Define fuzzy membership function as

\begin{matrix} μ_{F_{i}^{1}} (z_{i}) = \exp [\frac{- 0.5 (z_{i} + {1)}^{2}}{4}], \\ μ_{F_{i}^{2}} (z_{i}) = \exp [\frac{- 0.5 (z_{i} + {0.5)}^{2}}{4}], μ_{F_{i}^{3}} (z_{i}) = \exp [\frac{- 0.5 (z_{i})^{2}}{4}], \end{matrix}

\begin{matrix} μ_{F_{i}^{4}} (z_{i}) = \exp [\frac{- 0.5 (z_{i} - {0.5)}^{2}}{4}], \\ μ_{F_{i}^{5}} (z_{i}) = \exp [\frac{- 0.5 (z_{i} - {1)}^{2}}{4}], i = 1, \dots, 4 \end{matrix}

Suppose $B_{1} = [0.75, 0.75, 0.75]^{T}$ in (1), the initial value of the state is taken to be $x (0) = [1, 2, 0]^{T}$ . In simulation, we take $ξ_{1} (t)$ and $ξ_{2} (t)$ as band-limited white noises. By placing the poles $J$ at [-6 -4.2], we can obtain

L = [\begin{matrix} - 0.2229 & - 0.0743 & - 0.1486 \\ 0.2109 & 0.0703 & 0.1406 \end{matrix}] .

According to Theorem 1, it can be solved that

\begin{matrix} K = [\begin{matrix} - 6.2493 & - 1.2056 & - 8.1723 \end{matrix}], \\ R = [\begin{matrix} - 18.0888 & - 7.6857 & - 3.3239 \end{matrix}], \end{matrix}

\begin{matrix} Q_{1} = [\begin{matrix} 3.7992 & 1.7532 & - 0.9504 \\ 1.7532 & 1.5178 & - 0.6241 \\ - 0.9504 & - 0.6241 & 1.2256 \end{matrix}], \\ Q_{2} = [\begin{matrix} 1.0308 & - 0.3776 \\ - 0.3776 & 1.3584 \end{matrix}] . \end{matrix}

The simulation results are shown in Figure 2 and Table 1. Under the influence of multiple heterogeneous disturbances, the system without control is unstable as shown in Figure 2(a). Figure 2(b) displays that the states of the composite system (20) achieves asymptotically bounded in mean square. By comparison, it can be seen that the performance achieved by the proposed EADC strategy is satisfactory. Figure 2(c) expresses a curve of disturbance estimation error, which shows the feasibility of the ADO. The trajectory of parameters $θ$ can be illustrated in Figure 2(d).

Table 1

The anti-disturbance control accuracy under without control and EADC.

State	Fluctuation range without control	Fluctuation range under EADC
$x_{1}$	$3 ~ + \infty$	$3 ~ \pm 0.01$
$x_{2}$	$2 ~ + \infty$	$2 ~ \pm 0.05$
$x_{3}$	$- 1.5 ~ - \infty$	$- 1.5 ~ \pm 0.005$

As is shown in Figure 3 and Table 2, the anti-disturbance control accuracy of the EADC scheme is illustrated by comparing it with the classical control method $H_{\infty}$ control in simulations. The simulation results show that compared with $H_{\infty}$ control, the EADC scheme have the advantages of high-accuracy control performance and small system overshoot, which illustrate more satisfactory system performance can be achieved by the proposed EADC scheme.

Table 2.

The anti-disturbance control accuracy under $H_{\infty}$ control and EADC.

State	Fluctuation range under $H_{\infty}$ control	Fluctuation range under EADC
$x_{1}$	$- 12.5 ~ 12.5$	$- 0.08 ~ 0.08$
$x_{2}$	$- 3.5 ~ 3.5$	$- 0.05 ~ 0.05$
$x_{3}$	$- 2 ~ 2$	$- 0.01 ~ 0.01$

Performance analysis: (a) By placing the poles J at [-6; -4.2], the changes for any variation/combination /configuration of controller and disturbance parameters are shown in Tables 3–4.

The values of $W_{i}, V_{i}, F_{i}, L_{i}, K_{i}, (i = 1, 2, 3)$ are shown as follows

Parameters (1)

W_{1} = [\begin{matrix} - 1 & 5 \\ - 5 & - 4 \end{matrix}], V_{1} = [\begin{matrix} 5 & 0 \end{matrix}], F_{1} = [\begin{matrix} - 1.72 & 0.59 \end{matrix}],

\begin{matrix} L_{1} = [\begin{matrix} - 0.2229 & - 0.0743 & - 0.1486 \\ - 0.2109 & 0.0703 & 0.1406 \end{matrix}], \\ K_{1} = [\begin{matrix} - 6.2493 & - 1.2056 & - 8.1723 \end{matrix}] . \end{matrix}

Parameters (2)

W_{2} = [\begin{matrix} - 4 & - 10 \\ - 10 & - 8 \end{matrix}], V_{2} = [\begin{matrix} 8 & 0 \end{matrix}], F_{2} = [\begin{matrix} - 1.92 & 0.89 \end{matrix}],

\begin{matrix} L_{2} = [\begin{matrix} 0.0482 & - 0.0161 & 0.0321 \\ 0.2475 & 0.0825 & 0.1650 \end{matrix}], \\ K_{2} = [\begin{matrix} - 8.0095 & - 0.1130 & - 11.2501 \end{matrix}] . \end{matrix}

Parameters (3)

W_{3} = [\begin{matrix} - 8 & 15 \\ - 15 & - 10 \end{matrix}], V_{3} = [\begin{matrix} 10 & 0 \end{matrix}], F_{3} = [\begin{matrix} - 2.12 & 0.99 \end{matrix}],

\begin{matrix} L_{3} = [\begin{matrix} 0.1671 & 0.0557 & 0.1114 \\ 0.2883 & 0.0961 & 0.1922 \end{matrix}], \\ K_{3} = [\begin{matrix} - 12.9390 & 7.2392 & - 17.4492 \end{matrix}] . \end{matrix}

(b) Next, by placing the poles $J at [- 100, - 99]$ , the changes for any variation/combination/configuration of controller and disturbance parameters are shown as follows.

The values of $W_{i}, V_{i}, F_{i}, L_{j}, K_{j}, (i = 1, 2, 3 j = 4, 5, 6)$ are shown as follows

Parameters (4)

W_{1} = [\begin{matrix} - 1 & 5 \\ - 5 & - 4 \end{matrix}], V_{1} = [\begin{matrix} 5 & 0 \end{matrix}], F_{1} = [\begin{matrix} - 1.72 & 0.59 \end{matrix}],

\begin{matrix} L_{4} = [\begin{matrix} - 8.3143 & - 2.7714 & - 5.5429 \\ - 77.9571 & - 25.9857 & - 51.9714 \end{matrix}], \\ K_{4} = [\begin{matrix} - 57.1381 & - 128.7104 & - 73.3493 \end{matrix}] . \end{matrix}

Parameters (5)

W_{2} = [\begin{matrix} - 4 & - 10 \\ - 10 & - 8 \end{matrix}], V_{2} = [\begin{matrix} 8 & 0 \end{matrix}], F_{2} = [\begin{matrix} - 1.92 & 0.89 \end{matrix}],

\begin{matrix} L_{5} = [\begin{matrix} - 5.0893 & - 1.6964 & - 3.3929 \\ - 22.1571 & 7.3857 & 14.7714 \end{matrix}], \\ K_{5} = [\begin{matrix} - 29.6733 & 59.7565 & - 39.1348 \end{matrix}] . \end{matrix}

Parameters (6)

W_{3} = [\begin{matrix} - 8 & 15 \\ - 15 & - 10 \end{matrix}], V_{3} = [\begin{matrix} 10 & 0 \end{matrix}], F_{3} = [\begin{matrix} - 2.12 & 0.99 \end{matrix}],

\begin{matrix} L_{6} = [\begin{matrix} - 3.8786 & - 1.2929 & - 2.5857 \\ - 11.1214 & - 3.7071 & - 7.4143 \end{matrix}], \\ K_{6} = [\begin{matrix} - 22.1220 & 41.0133 & - 29.7214 \end{matrix}] . \end{matrix}

Based on the analysis of Table 3 and Table 4, the control gain $K$ and the observer gain $L$ will change accordingly with the changes of disturbance parameters. Table 3 and Table 4 demonstrate the robustness of the system under EADC is satisfactory and can be adjusted in different environments.

Table 3.

The variation/configuration of controller and disturbance parameters for system (1).

Parameters	Disturbance parameters	Disturbance observer	Controller
Parameters (1)	$W_{1}, V_{1}, F_{1}$	$L_{1}$	$K_{1}$
Parameters (2)	$W_{2}, V_{2}, F_{2}$	$L_{2}$	$K_{2}$
Parameters (3)	$W_{3}, V_{3}, F_{3}$	$L_{3}$	$K_{3}$

Table 4.

The variation/configuration of controller and disturbance parameters for system (1).

Parameters	Disturbance parameters	Disturbance observer	Controller
Parameters (1)	$W_{1}, V_{1}, F_{1}$	$L_{4}$	$K_{4}$
Parameters (2)	$W_{2}, V_{2}, F_{2}$	$L_{5}$	$K_{5}$
Parameters (3)	$W_{3}, V_{3}, F_{3}$	$L_{6}$	$K_{6}$

According to equation (37), the derivation process of upper bound composite system (20) is shown as follows

lim_{t \to \infty} \sup E | \bar{x} (t; t_{0}, {\bar{x}}_{0}) | ⩽ \frac{β}{μ λ_{max} (P)} = \frac{β}{α} .

(39)

Based on the work of Marzaki et al. (2015), the relevant performance of the system during the operation can be reflected by the indicators of integral of absolute error (IAE). If the value of IAE is smaller, the cumulative deviation of the system operation will be smaller, thus the damping and transient response characteristics of the system are better. In this paper, for the formula (31), there exists a constant $β ⩾ 0$ , such that $0 ⩽ γ (t) ⩽ β$ , where $γ (t) = α \bar{ε} (z) \bar{ε} (z)^{T} + Tr ({\bar{B}}_{1}^{T} P {\bar{B}}_{1}) + 2 [0 z (t)]^{T} P {\bar{F}}_{1} {\tilde{θ}}^{T} \hat{φ} (z)$ . The upper bound for the composite system (20) is $β$ . The system is evaluated by using IAE as follows

IAE = \int_{0}^{\infty} ∥ E (t) ∥ d (t) = \int_{0}^{\infty} ∥ \frac{β}{α} ∥ d (t) .

(40)

Since that $\bar{ε} (z)$ is a bounded error, $B_{1}^{T}$ and $P$ are bounded matrices, also considering that $z (t)$ , ${\bar{F}}_{1}$ , ${\tilde{θ}}^{T}$ and $\hat{φ} (z)$ are bounded matrices. Hence, when the $γ (t)$ is small enough then the boundary value $β$ will be small enough. Then, the value of IAE is smaller.

Application example and simulation

System (1) can describe a large class of practical applications in real world, such as the front-end speed controlled wind turbine (FSCWT) (Dong et al., 2019; Rubio et al., 2015). In the operation environment, the FSCWT is mainly driven by the stochastic uncertainty of natural wind. Then, the generator is easily affected by the external environment and internal multi-disturbance. Hence, these disturbances are mainly divided into two categories. One is the external disturbance that can be represented by the dynamic subsystem. The type of disturbances are accidental and stochastic, but their intensity is generally limited, which can be described as additive noise $ξ_{1} (t)$ and multiplicative noise $ξ_{2} (t)$ . The other is the internal disturbance with partially known information, which is introduced as the non-harmonic disturbance with unknown smooth nonlinear function.

Selecting the parameters of FSCWT, as in the work of Dong et al. (2013), the coefficient matrices of the state space mathematical model of the system (1) is described as follows

A = [\begin{matrix} 0.0057 & - 1.1498 & 0 & 0.0153 & 0.0164 & 0.6989 \\ - 7.9167 & 0.0396 & 0 & 7.5469 & 4.8125 & - 0.2685 \\ 0 & 0 & - 0.0633 & 0 & 0 & 0 \\ 0.0049 & - 0.9888 & 0 & 0.0165 & 0.0096 & 0.6011 \\ 0.0063 & - 0.8298 & 0 & - 0.0114 & 0.0127 & 0.7698 \\ - 6.8885 & - 0.0344 & 0 & 6.5668 & 6.5668 & - 0.2894 \end{matrix}],

B_{0} = [\begin{matrix} 0.7564 & 0 & 0 & - 0.6503 & - 0.8331 & 0 \\ 0 & 5.2083 & 0 & 0 & 0 & 4.5319 \\ 0 & 0 & 8.3333 & 0 & 0 & 0 \\ 0.6505 & 0 & 0 & - 0.7014 & 0.4879 & 0 \\ 0.6505 & 0 & 0 & 0.4879 & - 0.6456 & 0 \\ 0 & - 4.5319 & 0 & 0 & 0 & 4.8851 \end{matrix}],

B_{2} = [\begin{matrix} 0.0900 & 0.0300 & 0.0020 & - 0.0030 & 1.0300 & 0 \\ 0.0020 & 0.0400 & 0.0700 & - 0.0034 & 0.0034 & 0.0010 \\ - 0.0040 & - 0.0400 & - 0.0300 & - 0.0010 & - 0.0200 & - 0.0020 \\ - 0.0004 & - 0.0140 & 1.0000 & - 0.0080 & 0.1000 & 0 \\ 0.0400 & 0.0020 & 0.0070 & - 0.0340 & 0.0043 & 0.0010 \\ 0.0800 & 0.0200 & 0.0030 & 0.0040 & 1.0300 & 0 \end{matrix}] .

The non-harmonic disturbance $D_{0} (t)$ for the wind turbine is given by

\begin{matrix} W = [\begin{matrix} - 4 & 0 & 5 \\ 5 & - 5 & 1 \\ - 5 & 0 & - 4 \end{matrix}], V = {[\begin{matrix} 5 & 0 & 0 & - 5 & 0 & 5 \\ 0 & 0 & 5 & 0 & 0 & 0 \\ 0 & - 5 & 0 & 0 & 5 & 0 \end{matrix}]}^{T}, \\ F = [\begin{matrix} - 1.82 \\ - 2 \\ 2.99 \end{matrix}] . \end{matrix}

Suppose that $B_{1} = [0, 0.1, 0.1, 0.1, 0.1, 0]^{T}$ in (1), the initial value of the state is given to be $x (0) = [2.4415, 0.3322, 0, 3.0022, 0, 0.0704]^{T}$ . In simulation, we take $ξ_{1} (t), ξ_{2} (t)$ as band-limited white noises. By placing the poles $J$ at [-3 -3 -4], we can obtain

L = [\begin{matrix} 0.0215 & 0.1204 & 0.0000 & - 0.0055 & 0.0136 & - 0.0759 \\ - 0.0253 & - 0.0690 & 0.0480 & - 0.0307 & - 0.0002 & - 0.1248 \\ 0.0285 & 0.0895 & 0.0000 & 0.0285 & 0.0028 & 0.1055 \end{matrix}] .

Based on Theorem 1, it can be shown that

\begin{matrix} K & = [\begin{matrix} 12.9698 & - 3.0292 & - 2.7988 & - 0.6206 & - 8.6466 & - 2.9221 \\ 4.3290 & - 0.8420 & - 0.8204 & - 0.0193 & 0.5890 & - 0.5511 \\ 4.5075 & - 0.9079 & - 1.3888 & 0.0044 & - 0.9567 & - 0.8660 \\ 54.5955 & - 11.0191 & - 11.4436 & - 0.2902 & - 17.8971 & - 11.3646 \\ 62.1813 & - 12.2439 & - 12.8715 & - 0.5987 & - 12.3918 & - 13.3328 \\ 8.5772 & - 1.7775 & - 1.5149 & - 1.4946 & - 2.9312 & - 1.4937 \end{matrix}], \\ R & = [\begin{matrix} 3.6775 & - 19.4918 & 3.8740 & - 65.8832 & - 87.8663 & - 5.7570 \\ 16.8191 & - 6.7769 & 1.5171 & - 1.4690 & 18.4777 & 32.0717 \\ 4.0899 & - 0.9213 & - 26.7930 & 1.5033 & - 1.6775 & 9.1541 \\ 85.4704 & 4.2821 & 3.2652 & - 16.0323 & - 116.6727 & 26.3051 \\ 104.5069 & 0.7110 & 1.0238 & - 51.8662 & - 19.6552 & - 23.9326 \\ 20.1911 & - 7.2019 & 16.0770 & - 168.2117 & - 20.1976 & 31.7499 \end{matrix}], \\ Q_{1} & = [\begin{matrix} 11.5191 & 16.8367 & 11.0048 & 0.6617 & 0.9684 & 18.8687 \\ 16.8367 & 96.2963 & - 2.2885 & 1.4925 & - 4.3011 & - 3.8223 \\ 11.0048 & - 2.2885 & 58.8522 & 0.5308 & - 1.5268 & - 2.0720 \\ 0.6617 & 1.4925 & 0.5308 & 114.5064 & - 0.4142 & 0.3359 \\ - 0.9684 & - 4.3011 & - 1.5268 & - 0.4142 & 14.2609 & - 1.8211 \\ 18.8687 & - 3.8223 & - 2.0720 & 0.3359 & - 1.8211 & 96.9826 \end{matrix}], \\ Q_{2} & = [\begin{matrix} 1.3352 & - 0.2160 & - 0.2344 \\ - 0.2160 & 1.0188 & - 0.2638 \\ - 0.2344 & - 0.2638 & 1.6693 \end{matrix}] . \end{matrix}

The feasibility of the proposed EADC scheme can be seen in Figure 4. The simulation results show that although there are multiple heterogeneous disturbances in the FSCWT system, satisfactory system responses and high-accuracy control performances can be achieved with the proposed EADC scheme. Furthermore, the estimation error is satisfactory for ADO. According to the Table 5, the state responses of the system are diverging in the absence of control, but it tends to be 0.05 under EADC, which demonstrate effectiveness of the proposed scheme.

Table 5.

The anti-disturbance control accuracy under without control and EADC.

State	Fluctuation range without control	Fluctuation range under EADC
$x_{1}$	$3 ~ + \infty$	$3 ~ \pm 0.01$
$x_{2}$	$1 ~ + \infty$	$1 ~ \pm 0.05$
$x_{3}$	$- 1.5 ~ - \infty$	$- 1.5 ~ \pm 0.005$
$x_{4}$	$- 1 ~ - \infty$	$- 1 ~ \pm 0.001$
$x_{5}$	$2 ~ + \infty$	$2 ~ \pm 0.005$
$x_{6}$	$- 2.5 ~ - \infty$	$- 2.5 ~ \pm 0.01$

Conclusion

A class of stochastic system with multiple heterogeneous disturbances has been investigated in this paper. An ADO has been constructed to estimate the non-harmonic disturbances with unknown smooth nonlinear function, which can be approximated by fuzzy logic systems. Based on the ADO, an EADC scheme has been proposed to guarantee that the composite system achieves asymptotically bounded in mean square. The works for further research will consider the disturbance with unknown frequency, unknown amplitude and unknown phase for stochastic systems.

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: The work is supported by National Science Foundation of China 61973149, 61803195, 61903173, 61903174 and Shandong Provincial Natural Science Foundation, China ZR2019BF016, ZR2019PF014, ZR2019PF006, ZR2017MF052.

ORCID iD

Xinjiang Wei

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