Nonlinear disturbance observer-based control for a class of discrete-time stochastic systems with multiple heterogenous disturbances

Abstract

In this paper, the nonlinear disturbance observer-based control (NDOBC) scheme is proposed for a class of discrete-time stochastic systems with multiple heterogenous disturbances, which include the non-harmonic disturbance and a sequence of random vectors. A nonlinear disturbance observer (NDO) is designed to estimate the non-harmonic disturbance, then the NDOBC scheme is proposed by combining DOBC with $H_{\infty}$ control, such that the composite system can achieves asymptotically mean-square bounded and asymptotically mean-square stable in different conditions. Finally, simulation results show the effectiveness of the proposed method.

Keywords

Discrete-time stochastic system multiple heterogenous disturbances nonlinear exosystem nonlinear disturbance observer-based control

Introduction

Disturbances exist widely in practical engineering; as a key anti-disturbance control strategy, disturbance observer-based control (DOBC) had received much attention (Chen and Chen, 2010a; Guo and Chen, 2005; Kim et al., 1999; Mallon et al., 2006; Sun et al., 2014; Yang et al., 2006, 2016). In recent years, DOBC had been extended to deal with a class of systems with multiple disturbances (Guo and Cao, 2014a, 2014b; Wen et al., 2014; Zhang et al., 2014). Also, composite hierarchical anti-disturbance control (CHADC) was proposed by Guo and Cao (2014a, 2014b), such as composite DOBC and PI control (Yang and Tsubakihara, 2008), composite DOBC and $H_{\infty}$ control (Wei et al., 2009), composite DOBC and TSM control (Chen and Chen, 2010b), composite DOBC and roust adaptive control (Guo and Wen, 2011). The advantage of CHADC is that it can reject the disturbance of inner loop and attenuate the outer loop ones respectively, which improves the control accuracy of the system.

DOBC scheme has been successfully used for stochastic systems (Li et al., 2018; Sun et al., 2018; Wei et al., 2016). The DOBC method was applied to stochastic system with multiple disturbances (Wei et al., 2016). By constructing the disturbance observer to estimate the disturbance with partially-known information, then a disturbance observer-based disturbance attenuation control(DOBDAC) scheme is proposed to guarantee the composite system achieved asymptotically bounded in mean square. A disturbance observer is designed to estimate the disturbances with partially known information (Sun et al., 2018), then a disturbance attenuation and rejection scheme is proposed for stochastic Markovian jump system such that the state of closed-loop system is asymptotically bounded. The composite anti-disturbance control scheme (Li et al., 2018) is proposed for nonlinear singular stochastic hybrid system to attenuate and eliminate the affection of multiple disturbances.

On the other hand, the discrete-time stochastic systems play an important role in signal analysis and processing. A class of discrete-time stochastic systems is considered which include nonlinearity and multiple disturbances (Wei and Sun, 2018). A stochastic disturbance observer is designed to estimate the disturbance with partially known information, then an elegant anti-disturbance control (EADC) scheme is proposed such that the disturbances can be rejected and attenuated. For a class of discrete-time stochastic systems with nonlinearity and multiple disturbances, a CHADC scheme (Wei et al., 2018) combining disturbance observer and $H_{\infty}$ control is proposed to guaranteed the desired performances of the closed-loop system. In the above mentioned systems, the disturbance is generated by an exosystem, which represent the constant or harmonic disturbance. However, in practical engineering, non-harmonic disturbance exists widely, which has not received enough attention. Therefore, the research of the non-harmonic disturbance for stochastic systems is necessary.

In this paper, a class of discrete-time stochastic systems with multiple heterogenous disturbances is considered, the disturbances include the non-harmonic disturbance and a sequence of random vectors. The main contributions of this paper are as follows:

The disturbance includes two parts: one is the non-harmonic disturbance that is generated by an exosystem with a bounded nonlinear function, the other one is a sequence of random vectors including additive disturbance and multiplicative disturbance. Compared with Wei and Sun (2018), the range of disturbance has been extended.

A nonlinear disturbance observer (NDO) is proposed to estimate the non-harmonic disturbance, then a nonlinear disturbance observer-based control (NDOBC) scheme is proposed for a class of discrete-time stochastic systems by combining the NDO with $H_{\infty}$ control, such that the composite system is asymptotically mean-square bounded and asymptotically mean-square stable in different conditions.

The main content of this paper consists of the following parts. In section 2, the problem formulation is presented. In section 3, the main results are displayed, which include a NDO and a NDOBC scheme. In section 4, simulation examples are given to account for the availability of the strategy. In section 5, there is a summary of the paper.

Notations: The notation utilized in this paper is quite standard. The notation ∥.∥ refers to the Euclidean norm for vector and induced 2-norm for matrices. $E$ is the mathematical expectation operator. For a matrix $M, N$ , we denote by $M (N) < (\leq) 0$ to state that $M (N)$ is a negative (semi-) definite matrix. * represents the corresponding elements in the symmetric matrix.

Formulation of the problem

The following discrete-time stochastic system with multiple heterogenous disturbances is described as

\begin{matrix} x (k + 1) = A_{0} x (k) + B_{0} [u (k) + D_{0} (k)] + H d_{1} (k) \\ + B_{1} w_{1} (k) + B_{2} x (k) w_{2} (k), \end{matrix}

(1)

where $x (k) \in R^{n}$ , $u (k) \in R^{m} (m < n)$ are the system state and the control input, respectively. $d_{1} (k)$ is the process disturbance in the $H_{2} -$ norm. $A_{0} \in R^{n \times n}$ , and $B_{0} \in R^{n \times m}$ , $H \in R^{n \times m}$ , $B_{1} \in R^{n \times m}$ , $B_{1} \in R^{n \times m}$ , $B_{2} \in R^{n}$ are the coefficient matrices. $D_{0} (k) \in R^{m}$ represents the disturbance with partially known information, which is generate by the following exogenous system

{\begin{matrix} D_{0} (k) = D_{01} (k) + D_{02} (k), \\ D_{01} (k) = Vz (k), z (k + 1) = Wz (k) + Ff (z (k)), \\ D_{02} (k) = S w_{3} (k), \end{matrix}

(2)

where $V, W, F, S$ are known constant matrices. $f (z (k))$ is a known bounded nonlinear function. $w_{1} (k)$ , $w_{2} (k)$ and $w_{3} (k)$ are distributed random vectors defined on the complete probability space $(Ω,$ $F$ $, P)$ .

Remark 1. This formulation can describe the front-end speed controlled wind turbine (FSCWT), which is put forward by German VOITH company in recent years (Dong et al., 2013). In the operation environment of FSCWT, the state vector $x (k)$ represents current, and rotational speed, $d_{1} (k)$ represents the perturbations of uncertainties and the random vectors $w_{1} (k)$ , $w_{2} (k)$ represent the random wind turbine vibration that are accidental, random and their intensity is generally limited. $D_{0} (k)$ includes $D_{01} (k)$ and $D_{02} (k)$ represents the disturbances in the input channel of the wind turbine which is not matched with the control frequency of the wind turbine. $D_{01} (k)$ represents harmonic signal model of the nonlinear and $D_{02} (k)$ represents the random wind turbine vibration. The related parameters of FSCWT are shown in Dong et al. (2018), Rubio et al. (2015) and Sloth et al. (2009).

Substituting (2) into (1), yields

\begin{matrix} x (k + 1) = A_{0} x (k) + B_{0} u (k) + B_{0} Vz (k) + H d_{1} (k) \\ + B_{1} w_{1} (k) + B_{2} x (k) w_{2} (k) + B_{0} S w_{3} (k) . \end{matrix}

(3)

Letting $ζ (k) = {[\begin{matrix} w_{1}^{T} (k) w_{3}^{T} (k) \end{matrix}]}^{T}$ , $B = [\begin{matrix} B_{1} B_{0} S \end{matrix}]$ , then

\begin{matrix} x (k + 1) = A_{0} x (k) + B_{0} u (k) + B_{0} D_{01} (k) + B ζ (k) + H d_{1} (k) \\ + B_{2} x (k) w_{2} (k), \end{matrix}

(4)

D_{01} (k) = Vz (k), z (k + 1) = Wz (k) + Ff (z (k)) .

(5)

Remark 2. In reality, the disturbance model (2) can be used to describe a type of disturbances in engineering. In the case of $F = 0$ , the disturbances are restricted to unknown constant and harmonics with unknown phase and amplitude (Chen and Chen, 2010; Guo and Chen, 2005). In the case of $F \neq 0$ , the disturbances of most existing results are restricted to exosystems with non-harmonic dynamics (Lu and Huang, 2015; Wang et al. 2016).

Assumption 1. For any $z_{i} (k)$ (i=1, 2, ⋯), nonlinear functions $f (z_{i} (k))$ satisfy

∥ f (z_{1} (k)) - f (z_{2} (k)) ∥ \leq ∥ U (z_{1} (k) - z_{2} (k)) ∥,

(6)

where $U$ is the given matrix.

Assumption 2. $(A_{0}, B_{0})$ is controllable and the $(W, B_{0} V)$ is observable.

The following definition and lemmas for discrete-time stochastic systems are introduced by Kolmanovskii and Myshkis (1999), Feng et al. (2008) and Mao and Yuan (2006).

Definition 1. (Kolmanovskii and Myshkis, 1999) Consider the following discrete-time stochastic systems:

\begin{matrix} x (k + 1) = A_{0} x (k) + A_{1} x (k) w (k), \\ x (0) = x_{0} \in R^{n}, \end{matrix}

(7)

where $x (k) \in R^{n}$ is the system state, $A_{0} \in R^{n \times n}$ and $A_{1} \in R^{n \times n}$ are the coefficient matrices. $w (k)$ is a sequence of identically, independently normally distributed random vectors defined on the complete probability space $(Ω, F, P)$ . Then, system (7) is said to be asymptotically mean-square stable if

lim_{k \to \infty} E ∥ x (k) ∥^{2} = 0,

(8)

for all $x_{0} \in R^{n}$ .

Lemma 1. (Feng et al., 2008): System (7) is asymptotically mean-square stable if and only if there exists a matrix $P > 0$ such that

- P + A_{0}^{T} P A_{0} + A_{1}^{T} P A_{1} < 0 .

(9)

Lemma 2. (Mao and Yuan, 2006) consider the following discrete-time stochastic systems

x (k + 1) = f (x (k), k) + g (x (k), k) w (k),

(10)

with $x (t_{0}) = x_{0}$ . Let $f : R^{n} \to R^{n}$ , $g : R^{n} \to R^{m \times n}$ , $w (k)$ is a sequence of identically, independently normally distributed random vectors defined on the complete probability space $(Ω, F, P)$ and $a \lor b$ means the maximum of $a$ and $b$ . Assume that there exist two positive $K, K_{1}$ such that

(Lipschitz condition) for all $x, y \in R^{n}$

| f (x, k) - f (y, k) |^{2} \lor | g (x, k) - g (y, k) |^{2} \leq K | x - y |^{2};

(Linear growth condition) for all $(x, k) \in R^{n}$

| f (x, k) |^{2} \lor | g (x, k) |^{2} \leq K_{1} (1 + | x |^{2}) .

Then there exists a unique solution $x (k)$ to equation (10).

Main results

In this section, it is supposed that system states $x (k)$ are available. A NDO is designed to estimate the disturbance with partially known information $D_{0} (k)$ , then a composite control scheme is proposed.

NDO

The NDO is constructed as

{\begin{matrix} v (k + 1) = (W + L B_{0} V) \hat{z} (k) + L [A_{0} x (k) + B_{0} u (k)] + Ff (\hat{z} (k)), \\ {\hat{D}}_{01} (k) = V \hat{z} (k), \hat{z} (k) = v (k) - Lx (k), \end{matrix}

(11)

where $\hat{z} (k)$ is the estimation of $z (k)$ , and $v (k)$ is the state of the stochastic observer. $L$ is the gain of observer to be designed. The estimation error is denoted as $e_{z} (k + 1) = z (k) - \hat{z} (k)$ . Based on (4), (5) and (10), yields

\begin{matrix} e_{z} (k + 1) = (W + L B_{0} V) e_{z} (k) \\ + L [H d_{1} (k) + B ζ (k) + B_{2} x (k) w_{2} (k)] + F [f (z (k)) - f (\hat{z} (k))] . \end{matrix}

(12)

Since $(W, B_{0} V)$ is observable, the pole of error dynamics (11) can be placed at the left-hand side complex plane according to Zhang and Chen (2004) and Willems and Willems (1976). By adjusting $L$ , the performance requirement of NDO can be satisfied.

In the following, the disturbance observer-based controller is constructed as

u (k) = - {\hat{D}}_{01} (k) + Kx (k) .

(13)

Substituting (12) into (4), we have

\begin{matrix} x (k + 1) = (A_{0} + B_{0} K) x (k) + B_{0} V e_{z} (k) + B ζ (k) \\ + H d_{1} (k) + B_{2} x (k) w_{2} (k) . \end{matrix}

(14)

Combining (13) with (11), results in

\begin{matrix} [\begin{matrix} x (k + 1) \\ e_{z} (k + 1) \end{matrix}] = [\begin{matrix} A_{0} + B_{0} K & B_{0} V \\ 0 & W + L B_{0} V \end{matrix}] [\begin{matrix} x (k) \\ e_{z} (k) \end{matrix}] + [\begin{matrix} B \\ LB \end{matrix}] ζ (k) \\ + [\begin{matrix} B_{2} & 0 \\ L B_{2} & 0 \end{matrix}] [\begin{matrix} x (k) \\ e_{z} (k) \end{matrix}] w_{2} (k) + [\begin{matrix} H & 0 \\ LH & F \end{matrix}] [\begin{matrix} d_{1} (k) \\ f (z (k)) - f (\hat{z} (k)) \end{matrix}] . \end{matrix}

(15)

The composite system can be depicted by

\begin{matrix} \bar{x} (k + 1) = \bar{A} \bar{x} (k) + \bar{B} ζ (k) + {\bar{B}}_{2} \bar{x} (k) w_{2} (k) + \bar{F} d (k), \\ \bar{Z} (k) = T \bar{x} (k), \end{matrix}

(16)

where $\bar{Z} (k)$ is the reference output, $T = [\begin{matrix} T_{1} T_{2} \end{matrix}]$ is the weighting matrix and

\begin{matrix} \bar{x} (k) = [\begin{matrix} x (k) \\ e_{z} (k) \end{matrix}], \bar{A} = [\begin{matrix} A_{0} + B_{0} K & B_{0} V \\ 0 & W + L B_{0} V \end{matrix}], \bar{B} = [\begin{matrix} B \\ LB \end{matrix}], \\ {\bar{B}}_{2} = [\begin{matrix} B_{2} & 0 \\ L B_{2} & 0 \end{matrix}], \bar{F} = [\begin{matrix} H 0 \\ LH F \end{matrix}], d (k) = [\begin{matrix} d_{1} (k) \\ f (z (k)) - f (\hat{z} (k)) \end{matrix}] . \end{matrix}

(17)

NDOBC

In this section, we aim to design a NDOBC scheme, such that the state $\bar{x} (k)$ of the composite system (15) is asymptotically bounded in mean square. The following result can be obtained.

Theorem 1. For given discrete-time stochastic system (4) with disturbance (5) under Assumption 1–2, if there exist a constant $γ > 0$ , and matrices $Q_{1} > 0, Q_{2} > 0$ , and $R_{1}, L$ satisfying

Θ = [\begin{matrix} - Q_{1} & 0 & 0 & 0 & M_{1} & 0 & Q_{1} B_{2}^{T} & Q_{2} B_{2}^{T} L^{T} & Q_{1} T_{1}^{T} \\ * & - Q_{2} & 0 & 0 & Q_{2} V^{T} B_{0}^{T} & M_{2} & 0 & 0 & Q_{2} T_{2}^{T} \\ * & * & - γ^{2} I & 0 & H^{T} & 0 & 0 & 0 & 0 \\ * & * & * & - γ^{2} I & H^{T} L^{T} & F^{T} & 0 & 0 & 0 \\ * & * & * & * & - Q_{1} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - Q_{2} & 0 & 0 & 0 \\ * & * & * & * & * & * & - Q_{1} & 0 & 0 \\ * & * & * & * & * & * & * & - Q_{2} & 0 \\ * & * & * & * & * & * & * & * & - I \end{matrix}] < 0,

(18)

where

\begin{matrix} M_{1} = Q_{1} A_{0}^{T} + R_{1}^{T} B_{0}^{T}, \\ M_{2} = Q_{2} W^{T} + Q_{2}^{T} V^{T} B_{0}^{T} L^{T} . \end{matrix}

Then, by designing NDO (10) with gain $L$ and NDOBC (12) with gain $K = R_{1} Q_{1}^{- 1}$ , the composite system (15) is asymptotically mean-square bounded.

Proof: Consider the following Lyapunov function

V (\bar{x} (k), t) = {\bar{x}}^{T} (k) P \bar{x} (k) .

(19)

Letting

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

(20)

Choosing the following difference generator

L V (k) = E [V (x (k + 1), k + 1) - V (x (k), k)] .

(21)

(i) For $d (k) \neq 0$ , selecting $J_{N} = \sum_{k = 0}^{N} E ∥ \bar{Z} (k) ∥^{2} - γ^{2}$ $\sum_{k = 0}^{N} E ∥ d (k) ∥^{2}$ , then it can be verified that

\begin{matrix} J_{N_{1}} = E {\sum_{k = 0}^{N} {∥ \bar{Z} (k) ∥^{2} - γ^{2} ∥ d (k) ∥^{2} + L V (k)}} \\ - EV (x (N + 1)) \\ \leq E {\sum_{k = 0}^{N} {∥ \bar{Z} (k) ∥^{2} - γ^{2} ∥ d (k) ∥^{2} + L V (k)}} \end{matrix}

\begin{matrix} = E {\sum_{k = 0}^{N} {{\bar{x}}^{T} (k) T^{T} T \bar{x} (k) - γ^{2} d^{T} (k) d (k) + L V (k)}} \\ = E {\sum_{k = 0}^{N} {{\bar{x}}^{T} (k) T^{T} T \bar{x} (k) - γ^{2} d^{T} (k) d (k) \\ + {\bar{x}}^{T} (k + 1) P \bar{x} (k + 1) - {\bar{x}}^{T} (k) P \bar{x} (k)}} . \end{matrix}

(22)

Letting

η (k) = [\begin{matrix} \bar{x} (k) \\ \bar{d} (k) \end{matrix}], Θ_{1} = [\begin{matrix} Σ_{0} & {\bar{A}}^{T} P \bar{F} \\ {\bar{F}}^{T} P \bar{A} & {\bar{F}}^{T} P \bar{F} - γ^{2} I \end{matrix}],

(23)

Then

J_{N_{1}} \leq E {\sum_{k = 0}^{N} η^{T} (k) Θ_{1} η (k) + α} .

(24)

where $Θ_{0} = - P + {\bar{A}}^{T} P \bar{A} + {\bar{B}}_{2}^{T} P {\bar{B}}_{2}$ , $Σ_{0} = Θ_{0} + T^{T} T$ , and $α = E ({\bar{B}}^{T} P \bar{B})$ .

Since $\bar{B}$ and $P$ are bounded matrices, there exists a constant $β \geq 0$ , such that $0 \leq α \leq β$ . Based on Lemma 1, if $Θ_{1} < 0$ holds, then by designing NDO (10) with gain L and NDOBC (12) with gain $K = R_{1} Q_{1}^{- 1}$ , the composite system (15) is asymptotically mean-square bounded.

Next, we will proof $Θ_{1} < 0$ .

(1): $Θ_{1} < 0 \Leftrightarrow Θ_{2} < 0$ . Based on Schur complement, $Θ_{1} < 0$ is equivalent to $Θ_{2} < 0$ , where

Θ_{2} = [\begin{matrix} - P & 0 & {\bar{A}}^{T} P & {\bar{B}}_{2}^{T} P & T^{T} \\ * & - γ^{2} I & {\bar{F}}^{T} P & 0 & 0 \\ * & * & - P & 0 & 0 \\ * & * & * & - P & 0 \\ * & * & * & * & - I \end{matrix}] .

(25)

(2): $Θ_{2} < 0 \Leftrightarrow Θ_{3} < 0$ . Substituting (16) into (24), we can obtain $Θ_{2} < 0$ is equivalent to $Θ_{3} < 0$ , where

Θ_{3} = [\begin{matrix} - P_{1} & 0 & 0 & 0 & A_{0}^{T} P_{1} + K^{T} B_{0}^{T} P_{1} & 0 & B_{2}^{T} P_{1} & B_{2}^{T} L^{T} P_{2} & T_{1}^{T} \\ * & - P_{2} & 0 & 0 & V^{T} B_{0}^{T} P_{1} & W^{T} P_{2} + V^{T} B_{0}^{T} L^{T} P_{2} & 0 & 0 & T_{2}^{T} \\ * & * & - γ^{2} I & 0 & H P_{1} & 0 & 0 & 0 & 0 \\ * & * & * & - γ^{2} I & LH P_{1} & F^{T} P_{2} & 0 & 0 & 0 \\ * & * & * & * & - P_{1} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - P_{2} & 0 & 0 & 0 \\ * & * & * & * & * & * & - P_{1} & 0 & 0 \\ * & * & * & * & * & * & * & - P_{2} & 0 \\ * & * & * & * & * & * & * & * & - I \end{matrix}] .

(26)

(3): $Θ_{3} < 0 \Leftrightarrow Θ_{4} < 0$ . $Θ_{3} < 0$ is equivalent to $Θ_{4} < 0$ by pre-multiplied and post-multiplied diag ${Q_{1}, Q_{2}, I, I, Q_{1}, Q_{2}, Q_{1}, Q_{2}, I}$ simultaneously, where

Θ_{4} = [\begin{matrix} - Q_{1} & 0 & 0 & 0 & Q_{1} A_{0}^{T} + Q_{1} K^{T} B_{0}^{T} & 0 & Q_{1} B_{2}^{T} & Q_{2} B_{2}^{T} L^{T} & Q_{1} T_{1}^{T} \\ 0 & - Q_{2} & 0 & 0 & Q_{2} V^{T} B_{0}^{T} & Q_{2} W^{T} + Q_{2} V^{T} B_{0}^{T} L^{T} & 0 & 0 & Q_{2} T_{2}^{T} \\ * & * & - γ^{2} I & 0 & H^{T} & 0 & 0 & 0 & 0 \\ * & * & * & - γ^{2} I & H^{T} L^{T} & F^{T} & 0 & 0 & 0 \\ * & * & * & * & - Q_{1} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - Q_{2} & 0 & 0 & 0 \\ * & * & * & * & * & * & - Q_{1} & 0 & 0 \\ * & * & * & * & * & * & * & - Q_{2} & 0 \\ * & * & * & * & * & * & * & * & - I \end{matrix}] .

(27)

(4): $Θ_{4} < 0 \Leftrightarrow Θ < 0$ . Selecting $R_{1} = K Q_{1}$ , $Θ_{4} < 0$ is equivalent to $Θ_{1} < 0$ , means $Θ < 0$ .

According to Lemma 1, if $Θ < 0$ holds, the composite system (15) is asymptotically mean-square bounded for $d (k) \neq 0$ .

(ii) For $d (k) = 0$ , along with (15), we can obtain

\begin{matrix} L V (k) = E [{\bar{x}}^{T} (k + 1) P \bar{x} (k + 1) - {\bar{x}}^{T} (k) P \bar{x} (k)] \\ = E [(\bar{A} \bar{x} (k) + \bar{B} ζ (k) + {\bar{B}}_{2} \bar{x} (k) w_{2} (k))^{T} P (\bar{A} \bar{x} (k) + \bar{B} ζ (k) \\ + {\bar{B}}_{2} \bar{x} (k) w_{2} (k)) - {\bar{x}}^{T} (k) P \bar{x} (k)] \\ = E [{\bar{x}}^{T} (k) (- P + {\bar{A}}^{T} P \bar{A} + {\bar{B}}_{2}^{T} P {\bar{B}}_{2}) \bar{x} (k)] + E ({\bar{B}}^{T} P \bar{B}) \\ = E [{\bar{x}}^{T} (k) Θ_{0} \bar{x} (k)] + α . \end{matrix}

(28)

For system (15), based on Lemma 1, if $Θ_{0} < 0$ , by designing NDO (10) with gain L and NDOBC (12) with gain $K = R_{1} Q_{1}^{- 1}$ , the system is asymptotically mean-square bounded.

From (1)–(4), it can be seen that $Θ < 0 \Leftrightarrow Θ_{4} < 0 \Leftrightarrow Θ_{3} < 0 \Leftrightarrow Θ_{2} < 0 \Leftrightarrow Θ_{1} < 0$ , which guarantees $Θ_{0} < 0$ . Hence, the systen is asymptotically mean-square bounded for $d (k) = 0$ . The proof is completed.

In Theorem 1, a NDOBC scheme is proposed for a class of discrete-time stochastic systems with multiple heterogeneous disturbances, which include the non-harmonic disturbance with a nonlinear function and a sequence of random vectors. If there only exists multiplicative disturbance in system (4) with disturbance (5), the following result will be obtained.

Corollary 1. For given discrete-time stochastic system (4) with disturbance (5) under Assumption 1–2, for $B_{1} = 0$ , $S = 0$ , if there exist a constant $γ > 0$ , and matrices $Q_{1} > 0, Q_{2} > 0$ , and $R_{1}, L$ satisfying

Θ_{5} = [\begin{matrix} - Q_{1} & 0 & 0 & 0 & Q_{2} B_{2}^{T} L^{T} & Q_{1} T_{1}^{T} & Q_{1} B_{2}^{T} & N_{1} & 0 \\ * & - Q_{2} & 0 & 0 & 0 & Q_{2} T_{2}^{T} & 0 & Q_{2} V^{T} B_{0}^{T} & N_{2} \\ * & * & - γ^{2} I & 0 & 0 & 0 & 0 & H^{T} & 0 \\ * & * & * & - γ^{2} I & 0 & 0 & 0 & H^{T} L^{T} & F^{T} \\ * & * & * & * & - Q_{2} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - I & 0 & 0 & 0 \\ * & * & * & * & * & * & - Q_{1} & 0 & 0 \\ * & * & * & * & * & * & * & - Q_{1} & 0 \\ * & * & * & * & * & * & * & * & - Q_{2} \end{matrix}] < 0,

(29)

where

\begin{matrix} N_{1} = Q_{1} A_{0}^{T} + R_{1}^{T} B_{0}^{T}, \\ N_{2} = Q_{2} W^{T} + Q_{2}^{T} V^{T} B_{0}^{T} L^{T} . \end{matrix}

Then, by designing NDO (10) with gain $L$ and NDOBC (12) with gain $K = R_{1} Q_{1}^{- 1}$ , the composite system (15) is asymptotically mean-square stable.

Proof. Considering $B_{1} = 0$ , $S = 0$ in (4) and (5), according to Lemma 1, if $Θ_{5} < 0$ , then $LV (k) \leq E [{\bar{x}}^{T} (k) Θ_{5} \bar{x} (k)]$ holds, which means that the equilibrium $\bar{x} (k) = 0$ is asymptotically mean-square stable.

Simulation example

Consider the system (1) with coefficient matrices given in the work of Sathananthan et al. (2012),

\begin{matrix} A_{0} = [\begin{matrix} 1.1062 & 0.10901 \\ 0.40737 & 0.47394 \end{matrix}], B_{0} = [\begin{matrix} 0.34365 \\ 0.47573 \end{matrix}], \\ H = [\begin{matrix} 0.5 \\ 0.032 \end{matrix}], B_{2} = [\begin{matrix} 0.017005 & 0.10801 \\ 0.061999 & 0.14269 \end{matrix}] . \end{matrix}

The stochastic disturbance $D_{0} (k)$ in (2) is given by

W = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}], F = [\begin{matrix} - 0.1 \\ - 0.1 \end{matrix}], V = [\begin{matrix} 2 & 1 \end{matrix}] .

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

Suppose $B_{1} = [0.00990.0082; 0.00120.0022]$ , $S = [0.0020.001]$ , the nonlinear dynamic is denoted as $f (z (k)) = \sin (z_{1} (k)) * e^{2.3 (- z_{2} (k))}$ . Placing pole to $λ = [- 0.1, 0.1]$ , according to pole placement theorem, we can obtain

L = [\begin{matrix} - 0.2016 & - 0.2790 \\ 0.4031 & 0.5580 \end{matrix}] .

The initial value of the state is given to be $x (0) = [- 1, 1]^{T}$ . Based on Theorem 1, it can be solved that $γ = 10.2041$ and

\begin{matrix} Q_{1} = [\begin{matrix} - 0.7107 & 0.9328 \\ 0.9328 & 1.7766 \end{matrix}], R_{1} = [\begin{matrix} - 2.5645 & - 3.6817 \end{matrix}], \\ Q_{2} = [\begin{matrix} 0.5934 & 0.4100 \\ 0.4100 & 0.4705 \end{matrix}], K = [\begin{matrix} - 2.8587 & - 0.5714 \end{matrix}] . \end{matrix}

Figure 1 gives the comparison of the state of the closed-loop system with NDOBC, $H_{\infty}$ control and active disturbance rejection control (ADRC). And it shows the proposed NDOBC scheme has higher control accuracy compared with $H_{\infty}$ control and ADRC. Curves of the disturbance estimation error and the controller are given in Figures 2–3 respectively.

According to 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). When the value of IAE is smaller, the cumulative deviation of the system operation will be smaller, such that the damping and transient response characteristics of the system are better. The system is evaluated by using IAE as follows

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

(30)

In this paper, $\bar{B}$ and $P$ are bounded matrices, there exists a constant $β \geq 0$ , such that $0 \leq α \leq β$ , where $α = E ({\bar{B}}^{T} P \bar{B})$ . When $α$ is small enough then the boundary value $β$ will be small enough. Then, the value of IAE is smaller.

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

Suppose that $B_{1} = 0$ , $S = 0$ , the nonlinear dynamic is denoted by $f (z (k)) = \sin (z_{1} (k)) * e^{2.3 (- z_{2} (k))}$ . Placing pole to $λ = [- 0.3, 0.1]$ , according to pole placement theorem, we can obtain

L = [\begin{matrix} - 0.2854 & - 0.3950 \\ 0.3712 & 0.5138 \end{matrix}] .

Based on Corollary 1, it can be shown that $γ = 10.2041$ and

\begin{matrix} Q_{1} = [\begin{matrix} 0.7628 & 0.9787 \\ 0.9787 & 1.8363 \end{matrix}], R_{1} = [\begin{matrix} - 2.6238 & - 3.6751 \end{matrix}], \\ Q_{2} = [\begin{matrix} 0.5427 & 0.4250 \\ 0.4250 & 0.5246 \end{matrix}], K = [\begin{matrix} - 2.7577 & - 0.5316 \end{matrix}] . \end{matrix}

Figure 4 shows the comparison of the state of the closed-loop system with NDOBC, $H_{\infty}$ control and ADRC. It demonstrates the proposed NDOBC scheme can achieve more satisfying response compared with $H_{\infty}$ control and ADRC. Figure 5 gives the curve of disturbance estimation error and Figure 6 shows the curve of the controller.

Conclusion

A class of discrete-time stochastic systems with multiple heterogenous disturbances is considered in this paper, the disturbances include non-harmonic disturbance with a nonlinear function and a sequence of random vectors. A NDO is proposed to estimate the non-harmonic disturbance. Based on the NDO, a NDOBC scheme is proposed to guarantee the composite system asymptotically mean-square bounded. The difficulty is the coupling of the state and the disturbance estimate error, which results in the invalidity of certainty equivalence principle. In this paper, the composite pole placement and linear matrix inequality (LMI) methods are used to solve the difficulty.

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.

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Nonlinear disturbance observer-based control for a class of discrete-time stochastic systems with multiple heterogenous disturbances

Abstract

Keywords

Introduction

Formulation of the problem

Main results

NDO

NDOBC

Simulation example

For the case with B 1 ≠ 0 , S ≠ 0

For the case with B 1 = 0 , S = 0

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

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

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