Composite hierarchical anti-disturbance control for stochastic systems with multiple heterogeneous disturbances

Abstract

This paper studies the problem of anti-disturbance control for a class of stochastic systems with multiple heterogeneous disturbances, which include the white noise and the non-harmonic disturbance with unknown nonlinear function. An adaptive disturbance observer is constructed to estimate the non-harmonic disturbances with unknown nonlinear function, which is approximated by neural network. A composite hierarchical anti-disturbance control (CHADC) scheme is designed by integrated Lyapunov function and linear matrix inequality (LMI), such that the expected dynamic performance of the composite system is achieved. Finally, simulations show that the approach is proper and effective.

Keywords

Stochastic system multiple heterogeneous disturbances composite hierarchical anti-disturbance control neural network adaptive disturbance observer

Introduction

Disturbance widely exists in practical engineering, which may effect the control accuracy and even destroy the stability of the system. Therefore, anti-disturbance control has been a hot topic in control field. A number of advanced control strategies have emerged, such as active disturbance rejection control (ADRC) (Wang et al., 2018), adaptive control (Hu et al., 2019; Li and Wang, 2018), disturbance observer-based control (DOBC) (Guo and Wen, 2011; Liu et al., 2019; Wei and Chen, 2014; Zhang et al., 2016), sliding control (Liu et al., 2017; Wu et al., 2017; Zhao et al., 2019), filter design (Xu and Lam, 2007), $H_{\infty}$ control (Wei and Guo, 2010; Xu et al., 2017b), fuzzy control (Han et al., 2019b) and so on. As an important anti-disturbance control strategy, DOBC was founded in the late 1980s and has been successfully applied to various engineering systems (Back and Shim, 2008; Liu et al., 2013; Rong et al., 2015; Shima and Jo, 2009). Recently, DOBC has been extended to deal with the systems with multiple disturbances. Composite hierarchical anti-disturbance control (CHADC) strategies were put forward by Guo and Cao (2014a, 2014b) for systems with multiple disturbances. The CHADC structure can attenuate the outer loop disturbances and reject the inner loop ones separately and concurrently, which can achieve high anti-disturbance control accuracy.

On the other hand, the DOBC method was used for stochastic systems with multiple disturbances. The DOBC method was extended to stochastic systems with multiple disturbances, including disturbances generated by the linear exogenous system and white noise (Wei et al., 2016). In Sun et al. (2018), a composite DOBC and $H_{\infty}$ control was considered for stochastic Markovian jump systems with nonlinearity and multiple disturbances. Wei et al. (2018) proposed the CHADC scheme for a class of discrete time stochastic systems with with multiple disturbances and nonlinearity. The composite fault tolerant control (CFTC) with disturbance observer scheme was considered for a class of stochastic systems with faults and multiple disturbances (Sun et al., 2018). In Han et al. (2019a), for a class of stochastic systems, reduced-order observer based fault estimation and fault-tolerant control have been considered, and $H_{\infty}$ performance has been used to attenuate the disturbance.

Most of the aforementioned DOBC methods for stochastic systems only focus on the exogenous system with known parameter matrix. But, it is difficult to obtain all the exact parameter matrix in practical applications. Therefore, it is necessary to consider the exogenous system with unknown parameter matrix. Moreover, the modeling of nonlinear disturbance has been a difficult question in control areas. It is well known that neural network has been regarded as a powerful tool for highly approximate nonlinear dynamic systems. And it has been considered as a black box identifier, which can track nonlinear dynamics. Therefore, neural network method has been used for modeling of the nonlinear dynamics (Qiao et al., 2018; Wang et al., 2012; Yi et al., 2013). A novel incremental radial basis function (RBF) neural network was proposed for nonlinear systems modeling (Qiao et al., 2018). Recently, the combination of neural network with stochastic systems has received considerable attention (Chen et al., 2017; Gao et al., 2016; Wang et al., 2014). Adaptive neural control (ANC) was proposed for strict-feedback nonlinear stochastic systems (Chen et al., 2017). ANC was proposed for pure-feedback stochastic nonlinear systems (Wang et al., 2014). However, the existing current control methods for the stochastic system only focus on single disturbance or combine multiple disturbances into a new equivalent disturbance. Moreover, the sources and channels of actual disturbances are different, and the higher anti-disturbance control accuracy can be achieved by the comprehensive analyses about the influence mechanism for the multiple heterogeneous disturbances.

The purpose of this paper is to present a CHADC scheme for a class of stochastic system with multiple heterogeneous disturbances, which include the white noise and the non-harmonic disturbance with unknown nonlinear function. The main contributions of this paper are described as follows.

The current DOBC work for the stochastic system is extended to the CHADC work for a class of stochastic systems with multiple heterogeneous disturbances which include the white noise and the non-harmonic disturbance. The proposed CHADC scheme can make full use of the information of disturbances to achieve the higher anti-disturbance control accuracy.

The non-harmonic disturbance is modelled by an exogenous system with unknown nonlinear function, which is approximated by neural network. The range of the system disturbances is extended further. A new adaptive disturbance observer (ADO) is structured to estimate the non-harmonic disturbance by the approximation of the nonlinear function.

The rest of the paper is organised as follows. In Section 2, the problem formulation is introduced. An ADO and a CHADC are designed in Section 3. In Section 4, simulation examples are performed show that method is valid. In Section 5, a short summary of the proposed scheme is given.

Notations: For arbitrary matrix $M_{1}, M_{2}$ , $diag {M_{1}, M_{2}}$ stands for a block diagonal matrix, $M_{1} < 0$ means that $M_{1}$ is a negative definite matrix; we denoted by $λ_{\min}$ and $λ_{\max}$ to state that minimum and maximum of matrix eigenvalues, respectively; * represents the corresponding elements in the symmetric matrix; for a square matrix M, we denote $sym (M) = M + M^{T}$ ; $C^{2.1} (R^{n} \times R_{+})$ denotes the family of all nonnegative functions $V (x, t)$ on $R^{n} \times R_{+}$ which are $C^{2}$ in x and $C^{1}$ in t; $C^{k}$ denotes k-times continuously differentiable; $κ$ denotes the set of all functions: $(R_{+} \to R_{+})$ which are continuous, strictly in increasing and vanishing at zero; $κ_{\infty}$ denotes the set of all functions that are of class- $κ$ and unbounded.

Formulation of the problem

The stochastic system with multiple heterogeneous disturbances is modeled 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 $x (t) \in R^{n}$ represents the state and $u (t) \in R^{m} (m < n)$ represents the control input. The additive noise $ξ_{1} (t) \in R$ and multiplicative noise $ξ_{2} (t) \in R$ are band-limited white noises, which are considered as independently distributed random vectors, satisfy mathematical expectation $E {ξ_{1} (t)} = E {ξ_{2} (t)} = 0$ and variance $E | ξ_{1} (t) |^{2} = c_{1},$ $E | ξ_{2} (t) |^{2} = c_{2}$ , where $c_{1}$ and $c_{2}$ are constant. $A \in R^{n \times n}$ , $B_{0} \in R^{n \times m}$ , $B_{1} \in R^{n \times m}$ and $B_{2} \in R^{n}$ are the matrices of coefficient. $D_{0} (t) \in R^{m}$ represents the non-harmonic disturbance with unknown nonlinear function, which can be expressed by an exogenous system with suitable parameters

{\begin{matrix} \overset{\cdot}{z} (t) = Hz (t) + W^{*} σ (z (t)), \\ D_{0} (t) = Nz (t), \end{matrix}

(2)

where $z (t) \in R^{2 r}$ represent the state vector of the exogenous system. $W^{*}$ is the unknown matrix and $σ (z (t))$ is the known basis function of neural network that is designed to be Lipschitz continuous. $H, N$ are the known coefficient matrices.

Remark 1: Actually, a class of disturbance can be represented by the model of the disturbance (2) in practical engineering. For the case with $W^{*} = 0$ , the disturbances are restricted to unknown constants and harmonics with unknown phase and amplitude (Chen, 2004; Chen and Chen, 2010; Guo and Cao, 2014; etc.). For the case with $W^{*} \neq 0$ , the disturbances are restricted to non-harmonic disturbance with nonlinear function (Dong et al., 2019; Lu and Huang, 2015; Wang et al., 2017; etc.). Model (2) can be applied to the front-end control-type wind turbine (similar to Dong et al., 2019b; Rubio et al., 2015). The non-harmonic disturbance with nonlinear function $D_{0} (t)$ , which includes the constant value, harmonic noise and repetitive disturbances, because the typical disturbances that are not matched with the control frequency of the wind turbine are regarded as the disturbances in the input channel of the wind turbine.

According to Øksendal (2003), the substitution of $\frac{d W_{1} (t)}{dt}$ for $ξ_{1} (t)$ and $\frac{d W_{2} (t)}{dt}$ for $ξ_{2} (t)$ , one has

{\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) = Hz (t) dt + W^{*} σ (z (t)) dt, \\ D_{0} (t) = Nz (t), \end{matrix}

(3)

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.

Remark 2: According to Hu (2017) and Hu et al. (2014), filtration is used to model the information that is available at a given point and therefore play an important role in the formalization of stochastic processes. Let $(Ω, F, P)$ be a probability space and let I be an index set that is either ℕ, $R^{+}$ , or a subset of $R^{+}$ . For every $i \in I$ let $F_{i}$ be a sub $σ$ -algebra of F. Then

F_{t} : = (F_{i})_{i \in I}

is called a filtration, if $F_{k} \subseteq F_{ℓ} \subseteq F$ for all $k \leq ℓ$ . So filtration is families of $σ$ -algebra that is ordered non-decreasingly. If $F_{t}$ is a filtration, then $(Ω, F, F_{t}, P)$ is called a filtered probability space.

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

Assumption 2: By defining the estimation error as $e_{z} (t) = z (t) - \hat{z} (t)$ . $σ (z (t))$ is selected as the basis function of neural network, which satisfies the Lipschitz property

(σ (z (t)) - σ (\hat{z} (t)))^{T} (σ (z (t)) - σ (\hat{z} (t))) \leq e_{z}^{T} (t) U_{z} e_{z} (t),

(4)

where $U_{z}$ is a known positive definite matrix.

Assumption 3: $W^{*}$ is the unknown bounded matrix and satisfies

W^{*} (W^{*})^{T} \leq \bar{W},

(5)

where $\bar{W}$ is prior known positive definite matrix.

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

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

(6)

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 $f (0, t) = 0$ , $g (0, t) = 0$ . $B (t)$ , $t \geq 0$ 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}) \leq V (x, t), and LV (x, t) \leq - λ V (x, t) + β,

(7)

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

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

(8)

for all $(x, t) \in R^{n} \times R_{+} .$ That is, system (6) is asymptotically bounded in pth moment.

Lemma 2: (Deng and Krstic, 2001). For system (6), if there exist a $C^{2}$ function $V (x)$ , and class $κ_{\infty}$ functions $α_{1}$ and $α_{2}$ and class $κ$ functions $α_{3}$ , such that for all $x (t) \in R^{n}$ , $t \geq 0$

α_{1} (∥ x (t) ∥) \leq V (x, t) \leq α_{2} (∥ x (t) ∥),

(9)

\begin{matrix} LV (x, t) = \frac{\partial V}{\partial x} f (x, t) + \frac{1}{2} Tr {g (x, t)^{T} \frac{\partial^{2} V}{\partial x^{2}} g (x, t)} \\ \leq - α_{3} (∥ x (t) ∥) . \end{matrix}

(10)

Then, the equilibrium $x = 0$ is globally asymptotically stable in probability.

Main results

The system states $x (t)$ are supposed to be available. The non-harmonic disturbance is generated by an exogenous system with unknown nonlinear function, which is approximated by neural network. Based on the approximation of the nonlinear function, an ADO is constructed.

ADO

The ADO is structured to

{\begin{matrix} dv (t) = (H + L B_{0} N) \hat{z} (t) dt + \hat{W} σ (\hat{z} (t)) dt + L [Ax (t) + B_{0} u (t)] dt, \\ \hat{z} (t) = v (t) - Lx (t), \\ {\hat{D}}_{0} (t) = N \hat{z} (t), \end{matrix}

(11)

where the estimation of $D_{0} (t)$ is expressed by ${\hat{D}}_{0} (t)$ , and L is the observer gain to be determined. $v (t)$ is the auxiliary vector as the state of ADO. $\hat{W}$ is the estimation of the weight vector of neural network $W^{*}$ .

Denoting $\tilde{W} = W^{*} - \hat{W}$ . Based on (3) and (6), it is shown that the error dynamics satisfies

\begin{matrix} d e_{z} (t) = (H + L B_{0} N) e_{z} (t) dt + L B_{1} d W_{1} (t) + L B_{2} x (t) d W_{2} (t) \\ + W^{*} (σ (z (t)) - σ (\hat{z} (t))) dt + \tilde{W} σ (\hat{z} (t)) dt . \end{matrix}

(12)

According to Assumption 1, since $(H, B_{0} N)$ is observable, the poles of $d e_{z} (t) = (H + L B_{0} N) e_{z} (t) dt$ in (12) can be placed at the left-half side complex plane (Willems and Willems, 1976; Zhang and Chen, 2004). The performance requirement for ADO can be satisfied by adjusting L.

In the following, the CHADC is designed as

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

(13)

where K is the controller gain to be designed. Substituting (13) into (3), the closed-loop system is expressed as

\begin{matrix} dx (t) = (A + B_{0} K) x (t) dt + B_{0} N e_{z} (t) dt \\ + B_{1} d W_{1} (t) + B_{2} x (t) d W_{2} (t) . \end{matrix}

(14)

Combining (14) with (12), yields

\begin{matrix} [\begin{matrix} dx (t) \\ d e_{z} (t) \end{matrix}] = [\begin{matrix} A + B_{0} K & B_{0} N \\ 0 & H + L B_{0} N \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) + \tilde{W} σ (\hat{z} (t)) dt \\ + [\begin{matrix} B_{2} & 0 \\ L B_{2} & 0 \end{matrix}] x (t) d W_{2} (t) + W^{*} (σ (z (t)) - σ (\hat{z} (t))) dt, \end{matrix}

(15)

and 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) \\ + W^{*} (σ (z (t)) - σ (\hat{z} (t))) dt + \tilde{W} σ (\hat{z} (t)) dt, \end{matrix}

(16)

where

\begin{matrix} \bar{x} (t) = [\begin{matrix} x (t) \\ e_{z} (t) \end{matrix}], \bar{A} = [\begin{matrix} A + B_{0} K & B_{0} N \\ 0 & H + L B_{0} N \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}] . \end{matrix}

(17)

CHADC

The aims of this section is to design CHADC scheme such that the expected dynamic performance of the composite system is achieved, which is asymptotically bounded in mean square. Based on Lemma 1, the following result can be obtained.

Theorem 1: Considering stochastic system (3) with disturbance (2) under Assumptions 1–3, for given known matrices $U_{z}$ and $\bar{W}$ , if there exist matrices $Q_{1} = P_{1}^{- 1} > 0$ , $Q_{2} = P_{2}^{- 1} > 0$ , and $R_{1}$ , $R_{2}$ satisfying the following linear matrix inequality (LMI)

Ω_{1} = [\begin{matrix} Θ_{1} & Q_{1} B_{2}^{T} \\ * & - Q_{1} \end{matrix}] < 0,

(18)

Ω_{2} = [\begin{matrix} Θ_{2} & P_{2} \\ * & - {\bar{W}}^{- 1} \end{matrix}] < 0,

(19)

where

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

(20)

Θ_{2} = P_{2} H + H^{T} P_{2}^{T} + R_{2} B_{0} N + N^{T} B_{0}^{T} R_{2}^{T} + U_{z}^{T} U_{z},

(21)

and if the adaptive law satisfies

\overset{\cdot}{\hat{W}} = γ \hat{z} (t) P_{2} σ^{T} (\hat{z} (t)),

(22)

where $γ$ is designed positive constant. Then the composite system (16) is asymptotically bounded in mean square under the CHADC law (13) with gain $K = R_{1} Q_{1}^{- 1}$ and observer (11) with gain $L = P_{2}^{- 1} R_{2}$ .

Proof: For the stochastic system (16), consider the following Lyapunov function

V_{1} (x (t), t) = x^{T} (t) P_{1} x (t),

(23)

V_{2} (e_{z} (t), t) = e_{z}^{T} (t) P_{2} e_{z} (t) + Tr {{\tilde{W}}^{T} γ^{- 1} \tilde{W}} .

(24)

Selecting

\begin{matrix} Q_{1} = P_{1}^{- 1} > 0, \\ Q_{2} = P_{2}^{- 1} > 0 . \end{matrix}

(25)

Differentiating (23) along with (14), yields

\begin{matrix} L V_{1} (x (t), t) = x^{T} (t) [sym (P_{1} A + P_{1} B_{0} K)] x (t) + 2 x^{T} (t) P_{1} B_{0} N e_{z} (t) \\ + Tr (B_{1}^{T} P_{1} B_{1}) + Tr (x^{T} (t) B_{2}^{T} P_{1} B_{2} x (t)) \\ \leq x^{T} (t) [sym (P_{1} A + P_{1} B_{0} K)] x (t) \\ + 2 x^{T} (t) P_{1} B_{0} N e_{z} (t) \\ + Tr (B_{1}^{T} P_{1} B_{1}) + x^{T} (t) B_{2}^{T} P_{1} B_{2} x (t) \\ = x^{T} (t) [sym (P_{1} A + P_{1} B_{0} K) + B_{2}^{T} P_{1} B_{2}] x (t) \\ + 2 x^{T} (t) P_{1} B_{0} N e_{z} (t) + Tr (B_{1}^{T} P_{1} B_{1}) . \end{matrix}

(26)

Letting

\begin{matrix} Ω_{1} = sym (P_{1} A + P_{1} B_{0} K) + B_{2}^{T} P_{1} B_{2}, \\ ζ_{1} (t) = Tr (B_{1}^{T} P_{1} B_{1}), \end{matrix}

(27)

then

\begin{matrix} L V_{1} (x (t), t) \leq x^{T} (t) [sym (P_{1} A + P_{1} B_{0} K) + B_{2}^{T} P_{1} B_{2}] x (t) \\ + 2 x^{T} (t) P_{1} B_{0} N e_{z} (t) + Tr (B_{1}^{T} P_{1} B_{1}) \\ \leq x^{T} (t) Ω_{1} x (t) + 2 x^{T} (t) P_{1} B_{0} N e_{z} (t) + ζ_{1} (t) \\ \leq x^{T} (t) Ω_{1} x (t) + | η_{1} | ∥ x (t) ∥ ∥ e_{z} (t) ∥ + ζ_{1} (t) \\ \leq - | η_{2} | ∥ x (t) ∥^{2} + | η_{1} | ∥ x (t) ∥ ∥ e_{z} (t) ∥ + ζ_{1}, \end{matrix}

(28)

where $η_{1} > 0$ and $η_{2} > 0$ are proper constant. $ζ_{1} (t) = Tr (B_{1}^{T} P_{1} B_{1})$ corresponds to additive white noise in close-loop system. Considering that $B_{1}$ and P are bounded matrices, there exists a constant $ζ_{1} \geq 0$ , such that $0 \leq ζ_{1} (t) \leq ζ_{1}$ .

Based on (12) and Assumptions 2–3, the differential of (24) is presented as

\begin{matrix} L V_{2} (e_{z} (t), t) = e_{z}^{T} (t) [sym (P_{2} H + P_{2} L B_{0} N)] e_{z} (t) \\ + 2 e_{z}^{T} (t) P_{2} W^{*} (σ (z (t)) - σ (\hat{z} (t))) \\ + Tr (B_{1}^{T} L^{T} P_{1} L B_{1}) + Tr (x^{T} (t) B_{2}^{T} L^{T} P_{1} L B_{2} x (t)) \\ + 2 e_{z}^{T} (t) P_{2} \tilde{W} σ (\hat{z} (t)) + 2 Tr {{\overset{\cdot}{\tilde{W}}}^{T} γ^{- 1} \tilde{W}} \\ = e_{z}^{T} (t) [sym (P_{2} H + P_{2} L B_{0} N)] e_{z} (t) \\ + 2 e_{z}^{T} (t) P_{2} W^{*} (σ (z (t)) - σ (\hat{z} (t))) \\ + Tr (B_{1}^{T} L^{T} P_{1} L B_{1}) + Tr (x^{T} (t) B_{2}^{T} L^{T} P_{1} L B_{2} x (t)) \\ + 2 (z^{T} (t) - {\hat{z}}^{T} (t)) P_{2} \tilde{W} σ (\hat{z} (t)) + 2 Tr {{\overset{\cdot}{\tilde{W}}}^{T} γ^{- 1} \tilde{W}} \\ = e_{z}^{T} (t) [sym (P_{2} H + P_{2} L B_{0} N)] e_{z} (t) \\ + 2 e_{z}^{T} (t) P_{2} W^{*} (σ (z (t)) - σ (\hat{z} (t))) \\ + 2 z^{T} (t) P_{2} \tilde{W} σ (\hat{z} (t)) - 2 {\hat{z}}^{T} (t) P_{2} \tilde{W} σ (\hat{z} (t)) \\ + 2 Tr {{\overset{\cdot}{\tilde{W}}}^{T} γ^{- 1} \tilde{W}} \\ + Tr (B_{1}^{T} L^{T} P_{1} L B_{1}) + Tr (x^{T} (t) B_{2}^{T} L^{T} P_{1} L B_{2} x (t)) \\ \leq e_{z}^{T} (t) [sym (P_{2} H + P_{2} L B_{0} N) + P_{2} \bar{W} P_{2} \\ + U_{z}^{T} U_{z}] e_{z} (t) - 2 {\hat{z}}^{T} (t) P_{2} \tilde{W} σ (\hat{z} (t)) \\ + x^{T} (t) B_{2}^{T} L^{T} P_{1} L B_{2} x (t) + Tr (B_{1}^{T} L^{T} P_{1} L B_{1}) \\ - 2 Tr {{\tilde{W}}^{T} γ^{- 1} \overset{\cdot}{\hat{W}}} + 2 z^{T} (t) P_{2} \tilde{W} σ (\hat{z} (t)) . \end{matrix}

(29)

Selecting

\begin{matrix} Ω_{2} = sym (P_{2} H + P_{2} L B_{0} N) + P_{2} \bar{W} P_{2} + U_{z}^{T} U_{z}, \\ ζ_{2} (t) = ζ_{3} (t) + ζ_{4} (t) = Tr (B_{1}^{T} L^{T} P_{1} L B_{1}) + 2 z^{T} (t) P_{2} \tilde{W} σ (\hat{z} (t)), \end{matrix}

(30)

it can be obtained that

\begin{matrix} L V_{2} (e_{z} (t), t) \leq e_{z}^{T} (t) Ω_{2} e_{z} (t) + x^{T} (t) B_{2}^{T} L^{T} P_{1} L B_{2} x (t) \\ + Tr (B_{1}^{T} L^{T} P_{1} L B_{1}) + 2 z^{T} (t) P_{2} \tilde{W} σ (\hat{z} (t)) \\ \leq - | η_{3} | ∥ e_{z} (t) ∥^{2} + x^{T} (t) B_{2}^{T} L^{T} P_{1} L B_{2} x (t) \\ + ζ_{3} (t) + ζ_{4} (t) \\ \leq - | η_{3} | ∥ e_{z} (t) ∥^{2} + | η_{4} | ∥ x (t) ∥^{2} + ζ_{3} + ζ_{4} \\ = - | η_{3} | ∥ e_{z} (t) ∥^{2} + | η_{4} | ∥ x (t) ∥^{2} + ζ_{2}, \end{matrix}

(31)

where $η_{3} > 0$ and $η_{4} > 0$ are proper constant, and $| η_{1} |^{2} = | η_{2} | | η_{3} | - | η_{3} | | η_{4} |$ . $ζ_{3} (t) = Tr (B_{1}^{T} L^{T} P_{1} L B_{1})$ corresponds to additive white noise in error system. Considering that $B_{1}$ , L and P are bounded matrices, there exists a constant $ζ_{3} \geq 0$ , such that $0 \leq ζ_{3} (t) \leq ζ_{3}$ . Similar to above, considering that $z (t)$ , $\tilde{W}$ and $σ (\hat{z} (t))$ are bounded matrices, there exists a constant $ζ_{4} \geq 0$ , such that $0 \leq ζ_{4} (t) \leq ζ_{4}$ , and $ζ_{2} = ζ_{3} + ζ_{4}$ .

Based on (23) and (24), $V_{1}$ and $V_{2}$ are integrated into an integrated Lyapunov function for the composite system as follows

\begin{matrix} V (x (t), e_{z} (t), t) = V_{1} (x (t), t) + η_{0} V_{2} (e_{z} (t), t) \\ = x^{T} (t) P_{1} x (t) + η_{0} e_{z}^{T} (t) P_{2} e_{z} (t) + η_{0} Tr {{\tilde{W}}^{T} γ^{- 1} \tilde{W}}, \end{matrix}

(32)

where $η_{0}$ is a proper constant, and $η_{0} = \frac{η_{2} η_{3} - η_{1}^{2}}{η_{3} η_{4}}$ . According to (28), (31) and (32), we can obtain

\begin{matrix} LV (x (t), e_{z} (t), t) \leq (- | η_{2} | + | η_{4} | | η_{0} |) ∥ x (t) ∥^{2} \\ + | η_{1} | ∥ x (t) ∥ ∥ e_{z} (t) ∥ - | η_{0} | | η_{3} | ∥ e_{z} (t) ∥^{2} + η_{0} ζ_{2} + ζ_{1} \\ = {[\begin{matrix} x (t) \\ e_{z} (t) \end{matrix}]}^{T} [\begin{matrix} - | η_{2} | + | η_{4} | | η_{0} | & \frac{| η_{1} |}{2} \\ \frac{| η_{1} |}{2} & - | η_{3} | | η_{0} | \end{matrix}] [\begin{matrix} x (t) \\ e_{z} (t) \end{matrix}] + ζ \\ = - {\bar{x}}^{T} (t) P \bar{x} (t) + ζ \\ \leq - {\bar{x}}^{T} (t) λ_{\min} (P) \bar{x} (t) + ζ, \end{matrix}

(33)

where $ζ = η_{0} ζ_{2} + ζ_{1}$

and $P = [\begin{matrix} | η_{2} | - | η_{4} | | η_{0} | & \frac{- | η_{1} |}{2} \\ \frac{- | η_{1} |}{2} & | η_{3} | | η_{0} | \end{matrix}] .$ Obviously, P is positive matrix, such that

λ_{\min} (P) > 0 .

(34)

As a result, $LV (\bar{x} (t), t) \leq 0$ if inequality ${\bar{x}}^{T} (t) \bar{x} (t) \leq λ_{\min}^{- 1} (P) ζ$ holds. It can be shown that

{\bar{x}}^{T} (t) \bar{x} (t) \leq \max {{\bar{x}}_{0}^{T} (t) {\bar{x}}_{0} (t), λ_{\min}^{- 1} (P) ζ},

(35)

where ${\bar{x}}_{0} (t)$ is the initial value of $\bar{x} (t)$ . Based on (33) and (35), choosing $κ = λ_{\min} (P) | \bar{x} (t) |^{p}$ , $p = 2$ and $σ = \frac{α}{λ_{\max} (P)}$ , one has

κ (| \bar{x} (t) |^{p}) = λ_{\min} (P) | \bar{x} (t) |^{2} \leq V (\bar{x} (t), t),

(36)

LV (\bar{x} (t), t) \leq - σ V (\bar{x} (t), t) + ζ .

(37)

Based on Lemma 1, then

lim_{t \to \infty} supE | \bar{x} (t; t_{0}, x_{0}) |^{2} \leq κ^{- 1} (\frac{ζ}{σ}) .

(38)

Thus, the composite system (16) satisfies asymptotically bounded in mean square. The proof is completed.

Remark 3: According to Ikramov (2011) and John and Heike (2016), a congruence transformation is: A transformation of the form $A \to P^{T} AP$ , where A and P are square matrices, P is invertible, and $P^{T}$ denotes the transpose of P. In other words, two square matrices A and B over a field are called congruent if there exist an invertible matrix P over the same field such that $P^{T} AP = B$ where “T“ denotes the matrix transpose. Matrix congruence is an equivalence relation.

In Theorem 1, a CHADC is proposed for a class of stochastic systems with multiple heterogenous disturbances which include both multiplicative disturbance and additive disturbance. If there only exists multiplicative disturbance in the stochastic systems, the following result will be obtained.

Corollary 1: Considering stochastic system (3) with disturbance (2) under Assumptions 1–3, for $B_{1} = 0$ , $W^{*} = 0$ , if there exist matrices $Q_{1} = P_{1}^{- 1} > 0$ , $Q_{2} = P_{2}^{- 1} > 0$ , and $R_{1}$ , $R_{2}$ satisfying the following LMI

[\begin{matrix} Π_{1} & Q_{1} B_{2}^{T} \\ * & - Q_{1} \end{matrix}] < 0,

(39)

sym (P_{2} H + R_{2} B_{0} N) < 0,

(40)

where

Π_{1} = A Q_{1} + Q_{1}^{T} A^{T} + B_{0} R_{1} + R_{1}^{T} B_{0}^{T} .

(41)

Then the composite system (16) is globally asymptotically stable in probability under the CHADC law (13) with gain $K = R_{1} Q_{1}^{- 1}$ and observer (11) with gain $L = P_{2}^{- 1} R_{2}$ .

Proof: The process of proof can be given similarly to that of the proof for Theorem 1. Considering $B_{1} = 0, W^{*} = 0$ in (3), and class $κ_{\infty}$ function $α_{1} = λ_{\min} (P), α_{2} = λ_{\max} (P)$ , one has $λ_{\min} (P) | \bar{x} (t) |^{2} \leq V (\bar{x} (t), t) \leq λ_{\max} (P) | \bar{x} (t) |^{2}$ and $LV (\bar{x} (t), t) \leq - σ V (\bar{x} (t), t) .$ According to Lemma 2, the equilibrium $x (t) = 0$ is globally asymptotically stable in probability.

Simulation examples

Numerical simulation

In this section, to verify the result of the proposed method, a numerical example is given. To show the efficiency of the proposed method, a stochastic system in (1) is considered

A = [\begin{matrix} - 2 & 1 & 2 \\ 2 & - 5 & - 1 \\ 3 & 0.3 & - 1.2 \end{matrix}], B_{0} = [\begin{matrix} 2 \\ 1 \\ 2 \end{matrix}], B_{2} = [\begin{matrix} 0.1 & 0 & 0.1 \\ 0 & 1 & 0 \\ 0 & 0 & 0.2 \end{matrix}] .

The stochastic disturbance $D_{0} (t)$ is formulated by (2) with

H = [\begin{matrix} 0 & 22 \\ - 22 & 0 \end{matrix}], N = [\begin{matrix} 3.5 & 0 \end{matrix}] .

In addition, the basis function of neural network $σ (t) = [\begin{matrix} \arctan (t) \\ \sin (t) \end{matrix}]$ . For the exogenous system (2) with the adaptive laws (22), selecting $\bar{W} = [\begin{matrix} 2 & 0 \\ 0 & 2 \end{matrix}]$ . In the simulation, we take $ξ_{1} (t), ξ_{2} (t)$ as band-limited white noise.

For the case with $B_{1} \neq 0$

Suppose that $B_{1} = [0.2, 0.2, 0]^{T}$ in (1), the initial value of the state is taken to be $x (0) = [2, - 2, - 1]^{T}$ . According to Theorem 1, it can be solved that

\begin{matrix} L = [\begin{matrix} - 0.8164 & 0.1444 & - 0.2887 \\ - 0.3699 & - 0.3963 & - 0.7926 \end{matrix}], \\ K = [\begin{matrix} - 0.2638 & 0.8705 & - 0.5687 \end{matrix}], \\ R_{2} = [\begin{matrix} - 0.8139 & 0.1439 & 0.2878 \\ - 0.3687 & - 0.3951 & - 0.7902 \end{matrix}], \\ R_{1} = [\begin{matrix} - 0.2554 & 0.1070 & - 0.4628 \end{matrix}], \\ Q_{1} = [\begin{matrix} 0.4585 & 0.0987 & 0.3874 \\ 0.0987 & 0.1690 & 0.0247 \\ 0.3874 & 0.0247 & 0.6719 \end{matrix}], Q_{2} = [\begin{matrix} 1.0031 & 0 \\ 0 & 1.0031 \end{matrix}] . \end{matrix}

(42)

Comparing Figures 1a and 1b shows that the system state response curves from divergence to be asymptotically bounded, showing that the effectiveness of the controller is satisfactory. Figure 1c shows the disturbance estimation error, which illustrates estimation error of ADO is satisfying by using the proposed CHADC. Figure 1d illustrates the response of weight vector of neural network. The effectiveness of the proposed CHADC scheme can be seen in Figures 1a–1c the states of composite system (16) are asymptotically bounded in mean square.

The anti-disturbance control accuracy of the CHADC scheme is shown in Figure 2. The simulation result illustrates that the proposed CHADC scheme can achieve higher anti-disturbance control accuracy compared with $H_{\infty}$ control.

For the case with $B_{1} = 0$

Suppose that $B_{1} = [0, 0, 0]^{T}$ in (1), the initial value of the state is taken to be $x (0) = [2, - 2, - 1]^{T}$ . Based on Corollary 1, it can be obtained

\begin{matrix} L = [\begin{matrix} - 0.4230 & - 0.0023 & - 0.0045 \\ - 0.0071 & - 0.0035 & - 0.0071 \end{matrix}], \\ K = [\begin{matrix} - 0.2637 & 0.8702 & - 0.5687 \end{matrix}], \\ R_{2} = [\begin{matrix} - 0.3250 & - 0.0017 & - 0.0035 \\ - 0.0054 & - 0.0027 & - 0.0054 \end{matrix}], \\ R_{1} = [\begin{matrix} - 0.3313 & 0.1388 & - 0.6004 \end{matrix}], \\ Q_{1} = [\begin{matrix} 0.5949 & 0.1280 & 0.5025 \\ 0.1280 & 0.2192 & 0.0320 \\ 0.5025 & 0.0320 & 0.8716 \end{matrix}], Q_{2} = [\begin{matrix} 1.3013 & 0 \\ 0 & 1.3013 \end{matrix}] . \end{matrix}

(43)

Comparing Figure 3a and 3b shows that the system state response curves from divergence to be asymptotically bounded, showing that the effectiveness of the controller is satisfactory. Figure 3c is the curve of disturbance estimation error. The effectiveness of the proposed CHADC scheme can be demonstrated in Figure 3a-3c the states of composite system (16) are globally asymptotically stable in probability.

In Figure 4, the more satisfactory system robustness performance and higher anti-disturbance control accuracy can be achieved by comparing with CHADC scheme with $H_{\infty}$ control.

Application example and simulation

According to Dong et al. (2013, 2019a, 2019b), the coefficient matrix 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} 0.1567 & 0 & 0 & - 0.6503 & - 0.1031 & 0 \\ 0 & 5.1083 & 0 & 0 & 0 & 4.1019 \\ 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

W = [\begin{matrix} 0 & 0 & 5 \\ 0 & - 5 & 0 \\ - 5 & 0 & 0 \end{matrix}], V = {[\begin{matrix} 3.5 & 0 & 0 & - 8 & 0 & 3.5 \\ 0 & 0 & 3.5 & 0 & 0 & 0 \\ 0 & - 8 & 0 & 0 & 3.5 & 0 \end{matrix}]}^{T}

The initial value of the state is given to be $x (0) = [2, - 2, - 1, 1, - 0.5, - 1.5]^{T}$ . The basis function of neural network $σ (t) = [\begin{matrix} \arctan (t) \\ \sin (t) \\ - \arctan (t) \end{matrix}]$ . For the exogenous system (2) with the adaptive laws, selecting $\bar{W} = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}]$ . In simulation, we take $ξ_{1} (t), ξ_{2} (t)$ as band-limited white noises.

Suppose that $B_{1} = [0.2, 0, 0.2, 0, 0.2, 0]^{T}$ in (1) and (2). Based on Theorem 1, it can be shown that

L = [\begin{matrix} 2.4496 & 3.1679 & - 5.4674 & - 0.7352 & 4.13916 & - 44.1023 \\ - 6.0480 & - 0.0011 & - 68.2250 & - 17.5864 & 3.2560 & - 10.9501 \\ 6.4802 & 12.1707 & - 2.2502 & - 7.5720 & - 2.6936 & - 6.5491 \end{matrix}]

K = [\begin{matrix} - 6.9132 & 12.0074 & 0.4417 & - 25.4514 & 3.6690 & 13.8037 \\ 38.0799 & 1.2351 & - 1.6298 & - 7.0836 & - 15.9974 & 16.1984 \\ 0.0760 & 0.8403 & - 9.0851 & 0.04631 & 0.0867 & 0.1923 \\ - 46.2346 & 27.7513 & 0.0272 & - 20.0760 & - 4.7749 & 1.7869 \\ - 5.2033 & - 11.8343 & 0.9009 & - 0.6781 & 13.8433 & - 4.2750 \\ 16.5021 & - 7.0427 & - 0.7162 & - 40.8841 & - 54.2122 & - 0.8784 \end{matrix}]

R_{1} = [\begin{matrix} - 8.0341 & 13.4259 & 0.1130 & - 28.6582 & 1.5479 & 15.5556 \\ 43.6363 & 1.5938 & - 0.5209 & - 7.9196 & - 8.2072 & 18.2547 \\ 0.0508 & 0.8827 & - 5.1104 & 0.0092 & 0.1870 & 0.2173 \\ - 51.5688 & 31.3347 & 0.0536 & - 22.6640 & 0.6564 & 2.0118 \\ - 6.6719 & - 13.5263 & 0.1442 & - 0.7724 & 5.5704 & - 4.8166 \\ 21.8359 & - 6.9513 & 0.4542 & - 45.9882 & - 20.6304 & - 0.9910 \end{matrix}]

R_{2} = [\begin{matrix} 2.1734 & 2.8107 & - 4.8510 & - 0.6523 & 3.6724 & - 39.1259 \\ - 5.3660 & - 0.0010 & - 60.5322 & - 15.6034 & 2.8889 & - 9.7154 \\ 5.7495 & 10.7983 & - 1.9964 & - 6.7182 & - 2.3899 & - 5.8107 \end{matrix}]

Q_{1} = [\begin{matrix} 1.1200 & - 0.0019 & 0.0030 & 0.0015 & - 0.0628 & - 0.0001 \\ - 0.0019 & 1.1228 & 0.0065 & 0.0001 & - 0.0184 & - 0.0001 \\ 0.0030 & 0.0065 & 0.5630 & 0.0048 & - 0.0193 & - 0.0001 \\ 0.0015 & 0.0001 & 0.0048 & 1.1257 & - 0.0003 & 0.0000 \\ - 0.0628 & - 0.0184 & - 0.0193 & - 0.0003 & 0.3643 & - 0.0000 \\ - 0.0001 & - 0.0001 & - 0.0001 & 0.0000 & - 0.0000 & 1.1271 \end{matrix}]

Q_{2} = [\begin{matrix} 1.1271 & 0 & 0 \\ 0 & 1.1271 & 0 \\ 0 & 0 & 1.1271 \end{matrix}]

The feasibility of the proposed CHADC scheme can be seen in Figures 5 to 6. Comparing Figures 5a and 5b shows that system state response curves from divergence to be asymptotically bounded, showing that the effectiveness of the controller is satisfactory. Figure 5c gives the response of disturbance estimation error, and Figure 5d illustrates the response of weight vector of neural network. The simulation results shows that although multiple heterogeneous disturbances exist in the FSCWT system, the more satisfactory system robustness performance and higher anti-disturbance control accuracy can be achieved by comparing the $H_{\infty}$ control method. Furthermore, the estimation error is satisfactory for ADO by using the CHADC scheme.

Conclusion

In this paper, a class of stochastic systems with multiple heterogeneous disturbances has been investigated. For the non-harmonic disturbance with unknown nonlinear function, which is approximated by neural network, an ADO is constructed. Based on the approximation of the nonlinear function, the CHADC scheme is proposed such that composite system is 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 conflict of interests 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 61374108.

ORCID iD

Xinjiang Wei

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Composite hierarchical anti-disturbance control for stochastic systems with multiple heterogeneous disturbances

Abstract

Keywords

Introduction

Formulation of the problem

Main results

ADO

CHADC

Simulation examples

Numerical simulation

For the case with B 1 ≠ 0

For the case with B 1 = 0

Application example and simulation

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

For the case with $B_{1} \neq 0$

For the case with $B_{1} = 0$