Non-linear disturbance observer-based adaptive composite anti-disturbance control for non-linear systems with dynamic non-harmonic multisource disturbances

Abstract

In this paper, an adaptive composite anti-disturbance control structure is constructed for a class of non-linear systems with dynamic non-harmonic multisource disturbances. The key point of this paper is that a kind of non-harmonic disturbance, which has non-linear internal dynamics and complex features, is involved. A non-linear exogenous system is employed to describe the dynamic non-harmonic disturbances and several useful assumptions are introduced. By introducing a non-linear damping term, a novel adaptive non-linear disturbance observer is constructed. Based on the disturbance/uncertainty estimation and attenuation (DUEA) schemes, a composite anti-disturbance control structure is synthesized. Meanwhile, a new sufficient condition is derived and the stability of the closed-loop system is proved. Several illustrative examples are employed to demonstrate the effectiveness of the proposed method.

Keywords

Adaptive disturbance observers composite anti-disturbance control disturbance observer-based control disturbance/uncertainty estimation and attenuation non-harmonic multisource disturbance

Introduction

Disturbances and uncertainties exist more or less in industrial systems and may lead to control performance degradation and even instability of control systems (Chen et al., 2015, 2016; Gao, 2014). To achieve the desired control performance, rejection and attenuation for disturbances and uncertainties have received much attention over the past few years. A great many advanced control methods have been investigated for this issue, such as H_∞ control (Lavretsky and Wise, 2013; Wei, 1995), sliding mode control (Feng et al., 2013; Song et al., 2016; Wang, 2016), adaptive control (Gibson et al., 2015; Tao, 2014; Wang and Pan, 2017; Wang et al., 2017) and so on. In particular, the disturbance/uncertainty estimation and attenuation (DUEA) methods, including disturbance observer-based control (DOBC) (Chen et al., 2016; Li et al., 2014) and active disturbance rejection control (ADRC) (Guo, 2016; Jiang et al., 2015), have been studied recently (Chen et al., 2015, 2016). A feedback controller and a disturbance observer are designed independently and then integrated in DUEA methods; as a result, the balance between the nominal performance and robustness may be achieved from their two-degree-of-freedom structure (Chen et al., 2016). Benefiting from these advantages, DUEA methods may be widely used in a variety of practical engineering systems (An et al., 2016; Li and Liu, 2009; Yang et al., 2008).

In practice, disturbances and uncertainties usually derive from multiple sources. However, most of the DUEA results mentioned above are designed for single disturbances (Chen et al., 2015, 2016; Gao, 2014) and are not applicable to multisource disturbances. These multisource disturbances may have different features and should be described by a number of exogenous systems. Meanwhile, the complex characteristics of the multisource disturbances make it difficult to design the controllers. As a solution of the problem, based on DUEA structure, a composite hierarchical anti-disturbance control (CHADC) method has been reported (Guo and Cao, 2014; Sun and Guo, 2014; Wei and Chen, 2014; Wei et al., 2013; Yao and Guo, 2013). Aiming at constant and harmonic disturbances, a linear exogenous system is used for the description and a corresponding robust anti-disturbance controller is constructed. Additionally, for multisource disturbances with unknown frequency and amplitude, a modified composite hierarchical anti-disturbance controller has been investigated (Yang et al., 2016). By using the adaptive internal model principle and the observer backstepping technique, the stability of the closed-loop system is guaranteed.

In spite of these processes, most of the existing DUEA schemes mentioned above have never investigated dynamic non-harmonic disturbances. In recent research, dynamic disturbances are assumed to be harmonic or constant, and are formulated by a linear exogenous system (Guo and Cao, 2014; Li et al., 2016; Sun and Guo, 2014; Wei and Chen, 2014; Wei et al., 2013; Yang et al., 2016; Yao and Guo, 2013; Zong et al., 2016). Without considering the internal dynamics, several studies assume non-linear disturbances as a non-linear function (Sun and Guo, 2014; Wei and Chen, 2014; Wei et al., 2013) or bounded in the norm ||·||₂ (Yang et al., 2016; Yao and Guo, 2013). Therefore, it should be concluded that a kind of disturbance with non-linear internal dynamics and non-harmonic features, widely existing in practice, has never been considered. The complexity of the control design arises from the intrinsic complexity and non-linear characteristics of the dynamic non-harmonic disturbances. The existing anti-disturbance control schemes are no longer capable of achieving desired control performance.

Motivated by the aforementioned observations, this paper develops an adaptive composite anti-disturbance control for a class of non-linear systems with dynamic non-harmonic multisource disturbances. First, the dynamic non-harmonic disturbances are described by a non-linear exogenous system under several assumptions, whereas harmonic or constant disturbances are formulated by a linear system. Next, by introducing a non-linear damping term in the disturbance observer, an adaptive non-linear disturbance observer is constructed. Based on the DUEA schemes, an adaptive composite anti-disturbance control structure is finally established. Compared with the existing results, the main contributions of this paper are as follows: 1) to the best of the authors’ knowledge, it is the first time dynamic non-harmonic disturbances in DUEA structures are involved. A non-linear dynamic equation is employed to describe the dynamic non-harmonic disturbances and several useful assumptions are introduced. 2) A novel adaptive disturbance observer is proposed and a new sufficient condition is derived for the closed-loop stability. As a result, the estimation problem and the anti-disturbance control problem for systems with multisource dynamic non-harmonic disturbances are addressed. 3) The system states and the disturbance estimation errors may converge to an arbitrarily small value by choosing the design constants appropriately. Moreover, as an estimation and compensation method, the proposed control structure inherits the advantages of the DUEA schemes, without using conservative control gains.

The structure of the paper is organized as follows: in the next section, the control problem with dynamic non-harmonic multisource disturbances is formulated and some standard assumptions are introduced. The proposed adaptive composite anti-disturbance control structure is given, followed by stability analysis. Simulations are performed to verify the effectiveness of the proposed control approach.

Throughout this paper, the notations are defined as follows: $R^{n}$ denotes the real $n$ -dimensional space, while $R^{m \times n}$ denotes the space of $m \times n$ matrices with real entries. For a given matrix $A$ , $A^{T}$ denotes its transpose. The Euclidean norm is denoted $by ‖ \cdot ‖$ . Meanwhile, $I_{n}$ stands for $n$ -dimensional identity matrix.

Problem formulation

A class of continuous system with multisource disturbances is modelled as:

\overset{\cdot}{x} (t) = Ax (t) + Bu (t) + B_{1} d_{1} (t) + B_{2} d_{2} (t)

(1)

where $x (t) \in R^{n}$ is the state vector and $u (t) \in R^{m}$ is the control input. The disturbances and uncertainties under consideration are from multiple sources: $d_{1} (t) \in R^{r}$ denotes the constant and harmonic noises with partial known information; $d_{2} (t) \in R^{n}$ represents the external non-harmonic disturbance, which is described in Assumption 3. $A, B, B_{1}, B_{2}$ are matrices with proper dimensions. In this paper, we suppose that all of the system states are available.

The following assumptions are necessary:

Assumption 1. The harmonic disturbance $d_{1} (t)$ may be formulated by the following exogenous system:

\begin{matrix} d_{1} (t) = V_{1} ω_{1} (t) \\ {\overset{\cdot}{ω}}_{1} (t) = G_{1} ω_{1} (t) + H_{1} δ_{1} (t) \end{matrix}

(2)

where $G_{1} \in R^{r \times r}, H_{1} \in R^{r \times l_{1}}, V_{1} \in R^{r \times r}$ are known matrices. $δ_{1} (t) \in R^{l_{1}}$ is the additional disturbance from the perturbations and uncertainties in the exogenous system. It is supposed that $δ_{1} (t)$ is a bounded signal, e.g. $‖ δ_{1} (t) ‖ \leq δ_{1, max}$ .

Assumption 2. $(A, B)$ is controllable and $rank (B, B_{1}) = rank (B, B_{2}) = rank (B) = m$ .

Assumption 3. The non-harmonic disturbance $d_{2} (t)$ may be formulated by the following exogenous system:

\begin{matrix} d_{2} (t) = V_{2} ω_{2} (t) \\ {\overset{\cdot}{ω}}_{2} (t) = f (ω_{2} (t), x (t), u (t)) + H_{2} δ_{2} (t) \end{matrix}

(3)

where $H_{2} \in R^{n \times l_{2}}, V_{2} \in R^{n \times n}$ are known matrices. $δ_{2} (t) \in R^{l_{2}}$ is the additional disturbance in the dynamic system of $ω_{2} (t)$ . It is also supposed that $δ_{2} (t)$ is a bounded signal, e.g. $‖ δ_{2} (t) ‖ \leq δ_{2, max}$ . $f (ω_{2} (t), x (t), u (t)) \in R^{n}$ is an unknown non-linear function vector. There exists unknown constants $c_{1}, c_{2}, c_{3}, c_{4}$ such that

\begin{array}{l} ‖ f (ω_{2} (t), x (t), u (t)) ‖ \leq c_{1} + c_{2} ‖ ω_{2} (t) ‖ + c_{3} g_{1} (x (t), u (t)) \\ + c_{4} ‖ ω_{2} ‖ g_{2} (x (t), u (t)) \end{array}

(4)

where $g_{1} (x (t), u (t))$ and $g_{2} (x (t), u (t))$ are known non-negative functions.

Remark 1. Many practical control systems with multisource disturbances may be formulated in the form of (1), (2) and (3). The constant and harmonic disturbances may be represented by exogenous system (2) and a wide variety of non-harmonic disturbances may be represented by (3).

Remark 2. Assumption 1 presents the linear exogenous system for the constant and harmonic disturbances. Similar assumptions may be found in Sun and Guo (2014), Wei and Chen (2014), Wei et al. (2013), Yang et al. (2016), and Yao and Guo (2013). Guo and Cao (2014), Sun and Guo (2014), Wei and Chen (2014), Wei et al. (2013), Yang et al. (2016), Yao and Guo (2013) did not consider the internal dynamics; several studies assume the non-linear disturbances as a non-linear function (Sun and Guo, 2014; Wei and Chen, 2014; Wei et al., 2013) or bounded in the norm $‖ \cdot ‖_{2}$ (Yang et al., 2016; Yao and Guo, 2013). Formulating the non-harmonic disturbances as a non-linear exogenous system and introducing the constraints, Assumption 3 makes this paper, to the best of the authors’ knowledge, the first to involve the dynamic non-harmonic disturbances. In contrast to the disturbances in former results, $d_{2} (t)$ is not harmonic and has a non-linear exogenous system related to the system states, the control inputs and the additional disturbances, making it challenging to design controllers. Lastly, it should be noted that Assumption 2 ensures $d_{1}, d_{2}$ are in the range space of the control input u so that their effects may be compensated for through the control action.

The design object of this paper is summarized as follows:

Problem 1. Given the system (1) with dynamic non-harmonic multisource disturbances, it is required to design an anti-disturbance control structure to guarantee system stability in the presence of the multisource uncertainties.

The following lemma will be used in our design:

Lemma 1 (Jiang and Praly, 1998). For any $x$ and $y$ in $R^{n}$ , and for any positive real number $ρ$ , we have

x^{T} y = \frac{1}{2} ‖ x ‖^{2} + \frac{1}{2} ‖ y ‖^{2} - \frac{1}{2} ‖ x - y ‖^{2} \leq \frac{1}{4 ρ} ‖ x ‖^{2} + ρ ‖ y ‖^{2}

(5)

Adaptive composite anti-disturbance control

Control structure

The proposed adaptive composite anti-disturbance control structure comprises an adaptive non-linear disturbance observer and an anti-disturbance controller. The adaptive non-linear disturbance observer is constructed as

\begin{matrix} {\hat{d}}_{1} (t) = V_{1} {\hat{ω}}_{1} (t), {\hat{d}}_{2} (t) = V_{2} {\hat{ω}}_{2} (t), {\hat{ω}}_{1} (t) = z_{1} (t) - L_{1} x (t) \\ {\overset{\cdot}{\hat{ω}}}_{2} (t) = - L_{2} (z_{2} (t) - x (t)) - β (t) (z_{2} (t) - x (t)) φ ({\hat{ω}}_{2} (t), x (t), u (t)) \\ {\overset{\cdot}{z}}_{1} (t) = [G_{1} + L_{1} B_{1} V_{1}] {\hat{ω}}_{1} (t) + L_{1} [Ax (t) + Bu (t) + B_{2} V_{2} {\hat{ω}}_{2} (t)] \\ {\overset{\cdot}{z}}_{2} (t) = A z_{2} (t) + Bu (t) + B_{1} V_{1} {\hat{ω}}_{1} (t) + B_{2} V_{2} {\hat{ω}}_{2} (t) - L_{z} (z_{2} (t) - x (t)) \end{matrix}

(6)

where ${\hat{d}}_{1}, {\hat{d}}_{2}, {\hat{ω}}_{1}, {\hat{ω}}_{2}$ are the estimations of $d_{1} (t), d_{2} (t), ω_{1} (t), ω_{2} (t)$ , respectively. $L_{1}, L_{2}, L_{z}$ are the gain matrices of the disturbance observer. $z_{1} (t), z_{2} (t)$ are the auxiliary vector as the state of the observer. $φ ({\hat{ω}}_{2} (t), x (t), u (t))$ is a non-linear function, defined as:

\begin{matrix} φ ({\hat{ω}}_{2} (t), x (t), u (t)) = 2 + ‖ {\hat{ω}}_{2} (t) ‖^{2} + g_{1}^{2} (x (t), u (t)) \\ + g_{2}^{2} (x (t), u (t)) + ‖ {\hat{ω}}_{2} (t) ‖^{2} g_{2}^{2} (x (t), u (t)) \end{matrix}

(7)

$β (t)$ is an adaptive parameter with an adaptive law given by

\overset{\cdot}{β} (t) = η ({‖ z_{2} (t) - x (t) ‖}^{2} φ ({\hat{ω}}_{2} (t), x (t), u (t)) - σ β (t))

(8)

where $η > 0, σ > 0$ are design constants.

Based on the estimation of the disturbances, the anti-disturbance controller is designed as

u (t) = K_{0} x (t) - K_{1} {\hat{d}}_{1} (t) - K_{2} {\hat{d}}_{2} (t)

(9)

Stability analysis

Let $ε (t) = z_{2} (t) - x (t), {\tilde{ω}}_{1} (t) = {\hat{ω}}_{1} (t) - ω_{1} (t), {\tilde{ω}}_{2} (t) = {\hat{ω}}_{2} (t) - ω_{2} (t)$ , then from (1) and (3)–(6), we may obtain

\begin{array}{l} \dot{ε} (t) = A z_{2} (t) + B u (t) + B_{1} V_{1} {\hat{ω}}_{1} (t) + B_{2} V_{2} {\hat{ω}}_{2} (t) - L_{z} (z_{2} (t) - x (t)) \\ - [A x (t) + B u (t) + B_{1} d_{1} (t) + B_{2} d_{2} (t)] \\ = [A - L_{z}] ε (t) + B_{1} V_{1} {\tilde{ω}}_{1} (t) + B_{2} V_{2} {\tilde{ω}}_{2} (t) \\ {\dot{\tilde{ω}}}_{1} (t) = [G_{1} + L_{1} B_{1} V_{1}] {\hat{ω}}_{1} (t) + L_{1} [A x_{1} (t) + B u (t) + B_{2} V_{2} {\hat{ω}}_{2} (t)] \\ - L_{1} [A x (t) + B u (t) + B_{1} d_{1} (t) + B_{2} d_{2} (t)] - G_{1} ω_{1} (t) - H_{1} δ_{1} (t) \\ = [G_{1} + L_{1} B_{1} V_{1}] {\tilde{ω}}_{1} (t) + L_{1} B_{2} V_{2} {\tilde{ω}}_{2} (t) - H_{1} δ_{1} (t) \\ {\dot{\tilde{ω}}}_{2} (t) = - L_{2} ε (t) - β (t) ε (t) φ ({\hat{ω}}_{2} (t), \\ x (t), u (t)) - f (ω_{2} (t), x (t), u (t)) - H_{2} δ_{2} (t) \end{array}

(10)

By combining (1) and (9), the closed-loop system may be derived as

\dot{x} (t) = (A + B K_{0}) x (t) + (B_{1} - B K_{1}) d_{1} (t) - B K_{1} V_{1} {\tilde{ω}}_{1} (t) + (B_{2} - B K_{2}) d_{2} (t) - B K_{2} V_{2} {\tilde{ω}}_{2} (t)

(11)

As $rank (B, B_{1}) = rank (B, B_{2}) = rank (B) = m$ , the gain $K_{1}, K_{2}$ may be chosen such that $B_{1} - B K_{1} = 0$ and $B_{2} - B K_{2} = 0$ . Therefore, we may rewrite (11) as

\overset{\cdot}{x} (t) = (A + B K_{0}) x (t) - B K_{1} V_{1} {\tilde{ω}}_{1} (t) - B K_{2} V_{2} {\tilde{ω}}_{2} (t)

(12)

Defining $ξ (t) = {[x^{T} (t), ε^{T} (t), {\tilde{ω}}_{1}^{T} (t), {\tilde{ω}}_{2}^{T} (t)]}^{T}$ , the composite system yields

\begin{matrix} \overset{\cdot}{ξ} (t) = \bar{A} ξ (t) - \bar{B} f (ω_{2} (t), x (t), \\ u (t)) - β (t) \bar{B} ε (t) φ ({\hat{ω}}_{2} (t), x (t), u (t)) - H δ (t) \end{matrix}

(13)

where

\begin{matrix} \bar{A} = [\begin{matrix} A + B K_{0} & 0 & - B K_{1} V_{1} & - B K_{2} V_{2} \\ 0 & A - L_{z} & B_{1} V_{1} & B_{2} V_{2} \\ 0 & 0 & G_{1} + L_{1} B_{1} V_{1} & L_{1} B_{2} V_{2} \\ 0 & - L_{2} & 0 & 0 \end{matrix}], \bar{B} = [\begin{matrix} 0^{n \times n} \\ 0^{n \times n} \\ 0^{r \times n} \\ I^{n \times n} \end{matrix}] \\ H = [\begin{matrix} 0^{n \times n} \\ 0^{n \times n} \\ H_{1} \\ H_{2} \end{matrix}], δ (t) = [\begin{matrix} 0 \\ 0 \\ δ_{1} (t) \\ δ_{2} (t) \end{matrix}] \end{matrix}

(14)

Define $N = [0^{n \times n}, I^{n \times n}, 0^{n \times r}, 0^{n \times n}]$ , then $ε (t) = N ξ (t)$ .

Theorem 1. Consider system (1) with multisource disturbances (2) and (3) under Assumptions 1–3, if there exist positive-definite $Q, P$ satisfying

\begin{matrix} {\bar{A}}^{T} P + P \bar{A} = - Q \\ P \bar{B} = N^{T} \end{matrix}

(15)

then the non-linear disturbance observer given by (6)–(8) and the controller (9) may ensure that the system states and the disturbance estimation errors be uniformly bounded. Furthermore, the system states and the disturbance estimation error may converge to an arbitrarily small value by choosing the design constants $η, σ$ appropriately.

Proof. Consider the Lyapunov function candidate

V = \frac{1}{2} ξ^{T} (t) P ξ (t) + \frac{1}{2} η^{- 1} (β (t) - β^{*})^{2}

(16)

where $β^{*}$ is a constant value satisfying $β^{*} \geq max {c_{2} ρ_{1}, c_{4} ρ_{2}} / 2$ . $ρ_{1}$ and $ρ_{2}$ are positive constants that will be explained in the proof. Taking the derivative of $V$ with respect to time along (13), we may obtain

\begin{array}{l} \dot{V} = - \frac{1}{2} ξ^{T} (t) Q ξ (t) - ξ {(t)}^{T} P \bar{B} f (ω_{2} (t), x (t), \\ u (t)) - β (t) ξ^{T} (t) P \bar{B} ε (t) φ ({\hat{ω}}_{2} (t), x (t), u (t)) \\ - ξ^{T} (t) P \bar{B} H δ (t) + η^{- 1} (β (t) - β^{*}) \dot{β} (t) \end{array}

(17)

From Assumption 3 and (15) we come to

\begin{array}{l} \dot{V} \leq - \frac{1}{2} ξ^{T} (t) Q ξ (t) + ‖ ε (t) ‖ [c_{1} + c_{2} ‖ ω_{2} (t) ‖ + \\ c_{3} g_{1} (x (t), u (t)) + c_{4} ‖ ω_{2} (t) ‖ g_{2} (x (t), u (t))] \\ - β (t) ε^{T} (t) ε (t) φ ({\hat{ω}}_{2} (t), x (t), u (t)) \\ - ξ^{T} (t) P \bar{B} H δ (t) + η^{- 1} (β (t) - β^{*}) \dot{β} (t) \end{array}

(18)

By using $‖ ω_{2} (t) ‖ \leq ‖ {\hat{ω}}_{2} (t) ‖ + ‖ {\tilde{ω}}_{2} (t) ‖$ , it follows that

\begin{array}{l} \dot{V} \leq - \frac{1}{2} ξ^{T} (t) Q ξ (t) + ‖ ε (t) ‖ \\ [\begin{matrix} c_{1} + c_{2} ‖ {\hat{ω}}_{2} (t) ‖ + c_{2} ‖ {\tilde{ω}}_{2} (t) ‖ + c_{3} g_{1} (x (t), u (t)) \\ + c_{4} ‖ {\hat{ω}}_{2} (t) ‖ g_{2} (x (t), u (t)) + c_{4} ‖ {\tilde{ω}}_{2} (t) ‖ g_{2} (x (t), u (t)) \end{matrix}] \\ - β (t) ε^{T} (t) ε (t) φ ({\hat{ω}}_{2} (t), x (t), u (t)) \\ - ξ^{T} (t) P \bar{B} H δ (t) + η^{- 1} (β (t) - β^{*}) \dot{β} (t) \end{array}

(19)

According to Lemma 1, we know that for any $ρ_{1}, ρ_{2}, ρ_{3} > 0$ ,

\begin{matrix} c_{2} ‖ ε (t) ‖ ‖ {\tilde{ω}}_{2} (t) ‖ \leq \frac{1}{2} c_{2} (ρ_{1} {‖ ε (t) ‖}^{2} + \frac{1}{ρ_{1}} {‖ {\tilde{ω}}_{2} (t) ‖}^{2}) \\ c_{4} ‖ ε (t) ‖ ‖ {\tilde{ω}}_{2} (t) ‖ g_{2} (x (t), u (t)) \\ \leq \frac{1}{2} c_{4} (ρ_{2} {‖ ε (t) ‖}^{2} g_{2}^{2} (x (t), u (t)) + \frac{1}{ρ_{2}} {‖ {\tilde{ω}}_{2} (t) ‖}^{2}) \\ ‖ ξ^{T} (t) P \bar{B} H δ (t) ‖ \leq \frac{1}{2} (ρ_{3} {\bar{δ}}^{2} + \frac{ξ^{T} (t) ξ (t)}{ρ_{3}}) \end{matrix}

(20)

where $\bar{δ} \overset{Δ}{=}$ $sup_{t \geq 0} ‖ P \bar{B} H δ (t) ‖$ . Selecting $ρ_{1}, ρ_{2}$ such that $c_{2} / ρ_{1} + c_{4} / ρ_{2} \leq λ_{min} (Q) / 4$ and selecting $ρ_{3}$ such that $1 / ρ_{3} \leq λ_{min} (Q) / 4$ , we may obtain

\begin{array}{l} \dot{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) + ‖ ε (t) ‖ [c_{1} + c_{2} ‖ {\hat{ω}}_{2} (t) ‖ \\ + c_{3} g_{1} (x (t), u (t)) + c_{4} ‖ {\hat{ω}}_{2} (t) ‖ g_{2} (x (t), u (t))] \\ - β (t) ε^{T} (t) ε (t) φ ({\hat{ω}}_{2} (t), x (t), u (t)) + η^{- 1} (β (t) - β^{*}) \dot{β} (t) \\ + \frac{1}{2} c_{2} ρ_{1} {‖ ε (t) ‖}^{2} + \frac{1}{2} c_{4} ρ_{2} {‖ ε (t) ‖}^{2} g_{2}^{2} (x (t), u (t)) + \frac{1}{2} ρ_{3} {\bar{δ}}^{2} \end{array}

(21)

Substituting (7) and (8) into (21) yields

\begin{array}{l} \dot{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) + ‖ ε (t) ‖ [c_{1} + c_{2} ‖ {\hat{ω}}_{2} (t) ‖ \\ + c_{3} g_{1} (x (t), u (t)) + c_{4} ‖ {\hat{ω}}_{2} (t) ‖ g_{2} (x (t), u (t))] \\ - β^{*} ε^{T} (t) ε (t) [2 + {‖ {\hat{ω}}_{2} (t) ‖}^{2} + g_{1}^{2} (x (t), u (t)) \\ + g_{2}^{2} (x (t), u (t)) + {‖ {\hat{ω}}_{2} (t) ‖}^{2} g_{2}^{2} (x (t), u (t))] \\ - σ (β (t) - β^{*}) β (t) + \frac{1}{2} c_{2} ρ_{1} {‖ ε (t) ‖}^{2} \\ + \frac{1}{2} c_{4} ρ_{2} {‖ ε (t) ‖}^{2} g_{2}^{2} (x (t), u (t)) + \frac{1}{2} ρ_{3} {\bar{δ}}^{2} \end{array}

(22)

By using the Lemma 1 again, we obtain

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) - (β^{*} - \frac{1}{2} c_{2} ρ_{1}) ‖ ε (t) ‖^{2} \\ - (β^{*} - \frac{1}{2} c_{4} ρ_{2}) ‖ ε (t) ‖^{2} g_{2}^{2} (x (t), u (t)) \\ - σ (β (t) - β^{*}) β (t) + \frac{1}{4 β^{*}} ({c_{1}}^{2} + c_{2}^{2} + c_{3}^{2} + c_{4}^{2}) + \frac{1}{2} ρ_{3} {\bar{δ}}^{2} \end{matrix}

(23)

As $β^{*} \geq max {c_{2} ρ_{1}, c_{4} ρ_{2}} / 2$ and $- 2 (β (t) - β^{*}) β (t) \leq - {(β (t) - β^{*})}^{2} + {(β^{*})}^{2}$ , we may obtain

\overset{\cdot}{V} \leq - μ V + M

(24)

where

\begin{matrix} μ = min {\frac{λ_{min} (Q)}{2 λ_{max} (P)}, σ η} \\ M = \frac{1}{4 β^{*}} (c_{1}^{2} + c_{2}^{2} + c_{3}^{2} + c_{4}^{2}) + \frac{1}{2} ρ_{3} {\bar{δ}}^{2} + \frac{1}{2} σ {[β^{*}]}^{2} \end{matrix}

By solving (24) we may obtain

0 \leq V (t) \leq \frac{M}{μ} + (V (0) - \frac{M}{μ}) e^{- μ t}

(25)

It is obvious that $0 \leq V (t) \leq max {\frac{M}{μ}, V (0)}$ for all $t \geq 0$ , and the disturbance estimation errors and the system states are all bounded. Moreover, from (25), it is easy to find that the system states and the disturbance estimation errors may converge to an arbitrarily small value by choosing the design constants $η, σ$ appropriately. The proof is therefore complete. □

Simulations

In this section, we will investigate the effectiveness of the proposed control structure against dynamic non-harmonic multisource disturbances. Two cases of non-harmonic disturbances are taken into account. In Case 1, the non-linear exogenous system of the non-harmonic disturbance $d_{2} (t)$ is assumed to be

\begin{matrix} {\overset{\cdot}{ω}}_{2, 1} = ω_{2, 2} \\ {\overset{\cdot}{ω}}_{2, 2} = - \frac{4 ω_{2, 2}}{1 + e^{ω_{2, 2}}} + ω_{2, 2} \sin (ω_{2, 1} + ω_{2, 2}) \\ + 0.2 (ω_{2, 1} + ω_{2, 2}) (x_{1} x_{2} + \sin (0.1 u^{2})) + 0.2 (x_{1}^{2} + x_{2}^{2}) \cos (u) \end{matrix}

(26)

The coefficient matrices of $ω_{2}$ are assumed to be $V_{2} = [\begin{matrix} 1 & 1 \end{matrix}], H_{2} = {[\begin{matrix} 0.035 & 0.062 \end{matrix}]}^{T}$ . The harmonic disturbances $d_{1} (t)$ , which may be formulated by (2), have the following coefficient matrices:

G_{1} = [\begin{matrix} 0 & 0.5 \\ - 0.5 & 0 \end{matrix}], V_{1} = {[\begin{matrix} 7 \\ 3 \end{matrix}]}^{T}, H_{1} = [\begin{matrix} 0.022 \\ 0.015 \end{matrix}]

(27)

The following non-linear system with multi-source disturbances is under consideration

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = - 0.357 x_{1} + 0.68 x_{2} + d_{1} + d_{2} + u \end{matrix}

(28)

Moreover, we select the additional disturbances $δ_{1} (t), δ_{2} (t)$ as the sine wave signals with upper bound 1. The initial values of the system states and disturbances are taken to be $x (0) = {[2, 1]}^{T}$ , $ω_{1} (0) = {[1, 1]}^{T}$ and $ω_{2} (0) = {[10, 0]}^{T}$ . According to Theorem 1, the gain matrices are selected as

\begin{matrix} L_{1} = [\begin{matrix} 0 & 0.181 \\ 0 & - 2.4224 \end{matrix}], L_{2} = [\begin{matrix} - 5 & 0 \\ 0 & - 1 \end{matrix}], L_{z} = [\begin{matrix} - 5 & 0 \\ 0 & - 5.68 \end{matrix}] \\ K = [- 0.653, - 2.68], K_{1} = 1, K_{2} = 1 \end{matrix}

(29)

The non-linear damping term is selected as

\begin{matrix} φ ({\hat{ω}}_{2} (t), x (t), u (t)) = 2 + ‖ {\hat{ω}}_{2} (t) ‖^{2} + {[0.2 (x_{1}^{2} + x_{2}^{2}) \cos (u)]}^{2} \\ + 0.04 {[x_{1} x_{2} + \sin (0.1 u^{2})]}^{2} \\ + 0.04 ‖ {\hat{ω}}_{2} (t) ‖^{2} {[x_{1} x_{2} + \sin (0.1 u^{2})]}^{2} \end{matrix}

(30)

Meanwhile, we choose the following parameters: η = 0.01 and σ = 0.02. The simulation results are given in Figures 1 and 2.

In Case 2, the non-linear exogenous system of the non-harmonic disturbance $d_{2} (t)$ is taken as

\begin{matrix} {\overset{\cdot}{ω}}_{2, 1} = ω_{2, 2} \\ {\overset{\cdot}{ω}}_{2, 2} = \frac{- 0.05 (0.1 ω_{2, 1} + ω_{2, 2})}{1 + e^{ω_{2, 1} + ω_{2, 2}}} \\ + 0.2 (ω_{2, 1} + ω_{2, 2}) \cos (x_{1} + x_{2} + u + x_{1} x_{2}) \end{matrix}

(31)

The harmonic disturbances are assumed to have the same coefficient matrices as in Case 1. The non-linear system with multi-source disturbances (28) is still under consideration. The initial conditions and the gain matrices are the same. The non-linear damping term in Case 2 is selected as

\begin{array}{l} φ ({\hat{ω}}_{2} (t), x (t), u (t)) = 2 + {‖ {\hat{ω}}_{2} (t) ‖}^{2} \\ + \cos^{2} (x_{1} + x_{2} + u + x_{1} x_{2}) \\ + {‖ {\hat{ω}}_{2} (t) ‖}^{2} \cos^{2} (x_{1} + x_{2} + u + x_{1} x_{2}) \end{array}

(32)

The simulation results are given in Figures 3 and 4.

It may be found that the system states may converge to their desired value in both of the two cases, despite the dynamic non-harmonic multisource disturbances. The disturbance estimation errors are forced to converge to a small value, which may be the main reason for the precise stabilization. The effectiveness of the proposed method is verified. Moreover, from simulation results in the different situations, it may be concluded that the proposed method has robustness to different kinds of dynamic non-harmonic disturbances.

Conclusions

The anti-disturbance control problem for a class of non-linear systems with dynamic non-harmonic multisource disturbances is addressed in this paper. The dynamic non-harmonic disturbances are described by a non-linear exogenous system under several useful assumptions. By using a non-linear damping term, a novel adaptive non-linear disturbance observer is constructed. Based on the DUEA schemes, an adaptive composite anti-disturbance control structure is finally established. Meanwhile, a new sufficient condition is derived and stability of the closed-loop system is proved. Furthermore, the system states and the disturbance estimation error may converge to an arbitrarily small value by choosing the design constants appropriately. The simulation results demonstrate the effectiveness of the proposed method.

Footnotes

Declaration of conflicting interest

The authors declare that there is no conflict of interest.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grants no.11572248.

References

Liu

Wang

et al . (2016) Disturbance observer-based anti-windup control for air-breathing hypersonic vehicles. IEEE Transactions on Industrial Electronics PP(99): 1. https://doi.org/10.1109/TIE.2016.2516498

Chen

W-H

Ohnishi

Guo

(2015) Advances in disturbance/uncertainty estimation and attenuation [Guest editors’ introduction]. IEEE Transactions on Industrial Electronics 62(9): 5758–5762. https://doi.org/10.1109/TIE.2015.2453347

Chen

W-H

Yang

Guo

et al . (2016) Disturbance-observer-based control and related methods – an overview. IEEE Transactions on Industrial Electronics 63(2): 1083–1095. https://doi.org/10.1109/TIE.2015.2478397

Feng

Han

(2013) On nonsingular terminal sliding-mode control of nonlinear systems. Automatica 49(6): 1715–1722. https://doi.org/10.1016/j.automatica.2013.01.051

Gao

(2014) On the centrality of disturbance rejection in automatic control. ISA Transactions 53(4): 850–857. https://doi.org/10.1016/j.isatra.2013.09.012

Gibson

Annaswamy

et al . (2015) Adaptive output feedback based on closed-loop reference models. IEEE Transactions on Automatic Control 60(10): 2728–2733. https://doi.org/10.1109/TAC.2015.2405295

Guo

B-Z

(2016) Active disturbance rejection control: from ODEs to PDEs. IFAC-PapersOnLine 49(8): 278–283. https://doi.org/10.1016/j.ifacol.2016.07.454

Guo

Cao

(2014) Anti-disturbance control theory for systems with multiple disturbances: a survey. ISA Transactions 53(4): 846–849. https://doi.org/10.1016/j.isatra.2013.10.005

Jiang

Huang

Guo

(2015) Control of uncertain nonlinear systems based on observers and estimators. Automatica 59: 35–47. https://doi.org/10.1016/j.automatica.2015.06.012

10.

Jiang

Praly

(1998) Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties. Automatica 34(7): 825–840. https://doi.org/10.1016/S0005–1098(98)00018–1

11.

Lavretsky

Wise

(2013) Robust and Adaptive Control. London: Springer London. https://doi.org/10.1007/978–1–4471–4396–3

12.

Yang

Chen

W-H

et al . (2014) Disturbance Observer-based Control: Methods and Applications. Boca Raton, FL: CRC Press.

13.

Sun

Zong

et al . (2016) Composite anti-disturbance resilient control for Markovian jump nonlinear systems with partly unknown transition probabilities and multiple disturbances. International Journal of Robust and Nonlinear Control 18(October2014): 557–569. https://doi.org/10.1002/rnc.3682

14.

Liu

(2009) Adaptive speed control for permanent-magnet synchronous motor system with variations of load inertia. IEEE Transactions on Industrial Electronics 56(8): 3050–3059. https://doi.org/10.1109/TIE.2009.2024655

15.

Song

Niu

Zou

(2016) Finite-time sliding mode control synthesis under explicit output constraint. Automatica 65: 111–114. https://doi.org/10.1016/j.automatica.2015.11.037

16.

Sun

Guo

(2014) Composite adaptive disturbance observer based control and back-stepping method for nonlinear system with multiple mismatched disturbances. Journal of the Franklin Institute 351(2): 1027–1041. https://doi.org/10.1016/j.jfranklin.2013.10.002

17.

Tao

(2014) Multivariable adaptive control: a survey. Automatica 50(11): 2737–2764. https://doi.org/10.1016/j.automatica.2014.10.015

18.

Wang

(2016) Adaptive smooth second-order sliding mode control method with application to missile guidance. Transactions of the Institute of Measurement and Control 39(6): 848–860. https://doi.org/10.1177/0142331215621616

19.

Wang

Pan

(2017) Robust adaptive fault tolerant control for a class of nonlinear systems with dynamic uncertainties. Optik – International Journal for Light and Electron Optics 131: 941–952. https://doi.org/10.1016/j.ijleo.2016.11.209

20.

Wang

Yuan

Pan

et al . (2017) Adaptive neural control for high order Markovian jump nonlinear systems with unmodeled dynamics and dead zone inputs. Neurocomputing 247: 62–72. https://doi.org/10.1016/j.neucom.2017.03.041

21.

Wei

(1995) Nonlinear H/sub ∞/ control and its application to rigid spacecraft. IEEE Transactions on Automatic Control 40(7): 1281–1285. https://doi.org/10.1109/9.400476

22.

Wei

Chen

(2014) Composite hierarchical anti-disturbance control for nonlinear systems with DOBC and fuzzy control. International Journal of Robust and Nonlinear Control 24(2): 362–373. https://doi.org/10.1002/rnc.2891

23.

Wei

Chen

(2013) Composite adaptive disturbance observer-based control for a class of nonlinear systems with multisource disturbance. International Journal of Adaptive Control and Signal Processing 27(3): 199–208. https://doi.org/10.1002/acs.2290

24.

Yang

Ding

Chen

et al . (2016) Output-based disturbance rejection control for non-linear uncertain systems with unknown frequency disturbances using an observer backstepping approach. IET Control Theory & Applications 10(9): 1052–1060. https://doi.org/10.1049/iet-cta.2015.1160

25.

Yang

Z-J

Tsubakihara

Kanae

et al . (2008) A novel robust nonlinear motion controller with disturbance observer. IEEE Transactions on Control Systems Technology 16(1): 137–147. https://doi.org/10.1109/TCST.2007.903091

26.

Yao

Guo

(2013) Composite anti-disturbance control for Markovian jump nonlinear systems via disturbance observer. Automatica 49(8): 2538–2545. https://doi.org/10.1016/j.automatica.2013.05.002

27.

Zong

Hou

et al . (2016) Disturbance-observer-based-control and L 2−L ∞ resilient control for Markovian jump non-linear systems with multiple disturbances and its application to single robot arm system. IET Control Theory & Applications 10(2): 226–233. https://doi.org/10.1049/iet-cta.2015.0430