Nonlinear augmented state observer-based adaptive output feedback anti-disturbance control for nonlinear systems with non-harmonic multiple uncertainties

Abstract

In this paper, a novel output feedback anti-disturbance control method is carried out for a class of nonlinear systems subject to non-harmonic multisource disturbances. By using a nonlinear exogenous system, a class of non-harmonic disturbances possessing complex and nonlinear characteristics are taken fully into consideration. Based on a nonlinear damping term, we establish an adaptive augmented state observer that can achieve robust asymptotic disturbance estimation for the system states, the harmonic disturbances and the non-harmonic disturbances. By fusing the augmented state observer and a state-feedback controller, a robust adaptive output feedback anti-disturbance control structure is constructed. The boundness of the combined controller–observer system is derived on the basis of Lyapunov analysis. Furthermore, aiming at the intense non-harmonic disturbances, the proposed method is extended and a new output feedback controller is obtained. The effectiveness of the proposed scheme is demonstrated through experimental studies on a practical example.

Keywords

Adaptive control multisource uncertainties augmented state observer disturbance observer output feedback control

Introduction

Since the system uncertainties and external disturbances are difficult to be handled in a systematic way, the control design for the nonlinear systems becomes a challenging task. To achieve satisfactory control performance for the nonlinear systems, many advanced control design methods have been proposed, such as adaptive control (Chenliang and Yan, 2010; Tao, 2014; Wang and Lin, 2015), fuzzy control, neural network control (Bu et al., 2016; Wang et al., 2017b; Xu et al., 2016), fault tolerant control (Alwi and Edwards, 2008; Alwi et al., 2011; Fan et al., 2015; Wang et al., 2017a), and output feedback control (Lam and Li, 2013; Nguyen, 2018; Wei et al., 2017). Especially, when the system states are not available, the advantages of the output feedback controllers can be highlighted. For a class of discrete-time linear systems affected by bounded additive states and output disturbances, and subject to chance constraints, a stochastic output feedback control is proposed in Nguyen (2018). For a nonlinear descriptor system with actuator faults, a robust and reliable static output feedback (SOF) controller is designed in Wei et al. (2017). Moreover, in Du et al. (2014), an output-feedback finite-time synchronization structure is investigated for a class of second-order nonlinear multi-agent systems with a leader-follower architecture. In Li et al. (2017), an adaptive output-feedback controller with prescribed performance is designed for a class of switched nonlinear systems. Aiming at the polynomial fuzzy-model-based control system, Lam and Li (2013) have designed an output-feedback tracking controller.

On the other hand, because of the friction and load variation in mechanical and electrical systems, and the measurement noises or environment disturbances, the multi-source disturbances inevitably exist in most of the practical controlled processes. Therefore, to guarantee the system stability and high control accuracy, it is of significant importance to research the methods of attenuating and rejecting the disturbances. Among a variety of the disturbance attenuation and rejection control approaches, the disturbance observer-based control (DOBC) schemes and the disturbance/uncertainty estimation and attenuation (DUEA) controllers possess simple structure and powerful ability, and can be widely applied and easily implemented in a wealth of practical systems ( An et al., 2016; Chen, 2003). Chen et al. (2016) summarize the recent progresses of DOBC. As a further development, the multisource disturbances possessing different features have to be described by several exogenous systems, are classified and modeled in Guo and Cao (2014). For the purpose of handling the complex multisource disturbances, a novel composite hierarchical anti-disturbance control (CHADC) method has been proposed in Wei et al. (2013), Yao and Guo (2013), Sun and Guo (2014) and Wei and Chen (2014). However, the frequency and amplitude of the multisource disturbances are necessary. To overcome this limitation, a modified CHADC structure has been proposed by Yang et al. (2016).

However, it should be pointed that in the aforementioned DOBC results, the non-harmonic disturbances with dynamic features have never been investigated. In Wei et al. (2013), Yao and Guo (2013), Guo and Cao (2014), Sun and Guo (2014), Wei and Chen (2014), Li et al. (2016) and Yang et al. (2016), the disturbances are limited to be harmonic, and an auxiliary linear system is employed for description. The essential nonlinear characteristics of the non-harmonic disturbances are never taken into account (Li et al., 2016; Yang et al., 2016; Yao and Guo, 2013). The neglect of the widespread non-harmonic disturbances may induce control performance degradation or instability of the closed-loop system. Although fruitful results have been achieved on the DOBC and CHADC methods, the relative literature on the anti-disturbance control for the non-harmonic multiple disturbances has been really scattered, which motivates us to write this paper.

In this paper, a novel robust adaptive output feedback anti-disturbance controller is developed for a class of nonlinear systems subject to non-harmonic multisource disturbances. Because of the intrinsic non-harmonic characteristics, the control design problem becomes interesting and challenging. To estimate both of the system states and the multiple disturbances, a novel augmented state observer is established. By using an auxiliary nonlinear system, the non-harmonic multisource disturbances can be formulated and handled. Compared with the existing results, the major properties of the proposed control approach are summarized as follows:

As far as the author knows, it is the first time to consider the non-harmonic disturbances in the DOBC controllers. Hence, this paper has made progress for the DOBC methods.

Compared with Wei et al. (2013), Yao and Guo (2013, 2014) and Guo and Cao (2014), the considered disturbances are more general. Firstly, the disturbances are allowed to have nonlinear internal dynamics. Secondly, the assumptions of linear auxiliary system and the boundness in the norm $‖ \cdot ‖_{2}$ are removed.

The output feedback anti-disturbance control problem for the systems with non-harmonic multisource uncertainties can be addressed by integrating the augmented state observer and the anti-disturbance controller. Moreover, the results are extended to the non-harmonic disturbances with more intense nonlinearities.

The remainder of the paper is given as follows. In Section 2, the control problem with dynamic non-harmonic multisource disturbances and several standard assumptions are introduced. The proposed robust adaptive output feedback anti-disturbance control structure and the convergence analysis are provided in Section 3, followed by the extended results in Section 4. To verify the effectiveness of the proposed control approach, the simulations are performed in Section 5. 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 ‖·‖. $I_{n}$ stands for n -dimensional identity matrix.

Problem formulation

Consider the following continuous nonlinear system subject to multiple disturbances

\begin{array}{l} \overset{\cdot}{x} (t) = A_{0} x (t) + B_{0} u (t) + B_{1} d_{1} (t) + B_{2} d_{2} (t) \\ y (t) = C_{0} x (t) + C_{1} d_{1} (t) + C_{2} d_{2} (t) \end{array}

(1)

where $x (t) \in ℜ^{n}$ and $u (t) \in ℜ^{m}$ are the system states and the control inputs, respectively. $y \in ℜ^{p}$ is the system output vector. In this paper, we consider the disturbances from multiple sources. $A_{0}, B_{0}$ are the system matrices. $B_{1}, B_{2}$ are the disturbance distribution matrices. $C_{0}, C_{1}, C_{2}$ are the output matrices corresponding to the system states, the harmonic disturbances and the non-harmonic disturbances, respectively. $A_{0}, B_{0}, B_{1}, B_{2}, C_{0}, C_{1}, C_{2}$ are with proper dimensions. $d_{1} (t) \in ℜ^{r}$ represents the constant and harmonic noises with partial known information. The internal dynamics of $d_{1} (t)$ can be described by the following auxiliary 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 $δ_{1} (t) \in ℜ^{l_{1}}$ denotes the additional disturbance satisfying $‖ δ_{1} (t) ‖ \leq δ_{1, max}$ . $G_{1} \in ℜ^{r \times r}, V_{1} \in ℜ^{r \times r}, H_{1} \in ℜ^{r \times l_{1}}$ are known matrices. The external non-harmonic disturbance is described by $d_{2} (t) \in ℜ^{p}$ , which possesses the following dynamic equations

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

(3)

where $G_{2} \in ℜ^{p \times p}, V_{2} \in ℜ^{p \times p}, H_{2} \in ℜ^{p \times l_{2}}$ are known matrices, $δ_{2} (t) \in ℜ^{l_{2}}$ is the additional disturbance existing in (3). $f (ω_{2} (t), y (t), u (t)) \in ℜ^{p}$ is an unknown nonlinear function. $‖ δ_{2} (t) ‖ \leq δ_{2, max}$ .

Throughout this paper, we make following assumptions

Assumption 1: $rank (B_{0}, B_{1}) = rank (B_{0}, B_{2}) = rank (B_{0}) = m$ .

Assumption 2: There exist unknown constants $c_{1}, c_{2}, c_{3}, c_{4}$ such that

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

(4)

where $g_{1} (y, u)$ and $g_{2} (y, u)$ are known nonnegative functions.

The control objective of the paper is to develop an output feedback control structure to guarantee the closed-loop system convergence for system (1) subjected to both harmonic and non-harmonic disturbances.

Adaptive output feedback composite anti-disturbance control method

In this section, we suppose that Assumptions 1–3 hold. Noting that all states of the system are available, it is required to develop an observer to estimate both of the states and the disturbances. A robust adaptive augmented state observer is designed as

\begin{matrix} \overset{\cdot}{\hat{x}} (t) = A_{0} \hat{x} (t) + B_{0} u (t) + B_{1} V_{1} {\hat{ω}}_{1} (t) + B_{2} V_{2} {\hat{ω}}_{2} (t) \\ - L_{0} (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \\ {\hat{d}}_{1} (t) = V_{1} {\hat{ω}}_{1} (t), {\hat{d}}_{2} (t) = V_{2} {\hat{ω}}_{2} (t), \\ {\overset{\cdot}{\hat{ω}}}_{1} (t) = G_{1} {\hat{ω}}_{1} (t) - L_{1} (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \\ {\overset{\cdot}{\hat{ω}}}_{2} (t) = G_{2} {\hat{ω}}_{2} (t) - β (t) (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \\ φ ({\hat{ω}}_{2} (t), y (t), u (t)) \\ - L_{2} (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \end{matrix}

(5)

where $\hat{x} (t), {\hat{d}}_{1} (t), {\hat{d}}_{2} (t), {\hat{ω}}_{1} (t), {\hat{ω}}_{2} (t)$ are the estimates of $x (t), d_{1} (t), d_{2} (t), ω_{1} (t), ω_{2} (t)$ , respectively. $φ ({\hat{ω}}_{2} (t), y (t), u (t))$ is a nonlinear function formulated by

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

(6)

$L_{0}, L_{1}, L_{2}$ are the gain matrices of observer (5). $β (t)$ is an adaptive parameter. The update law for $β (t)$ is designed as

\overset{\cdot}{β} (t) = η ({‖ C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t) ‖}^{2} φ ({\hat{ω}}_{2} (t), y (t), u (t)) - σ β (t))

(7)

where $η, σ > 0$ . Accordingly, the control law is designed as

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

(8)

Define $\tilde{x} (t) = \hat{x} (t) - x (t), {\tilde{ω}}_{1} (t) = {\hat{ω}}_{1} (t) - ω_{1} (t), {\tilde{ω}}_{2} (t) = {\hat{ω}}_{2} (t) - ω_{2} (t)$ . Then from (1)–(5), it is easy to get that

\begin{matrix} \overset{\cdot}{\tilde{x}} (t) = A_{0} \hat{x} (t) + B_{0} u (t) + B_{1} V_{1} {\hat{ω}}_{1} (t) + B_{2} V_{2} {\hat{ω}}_{2} (t) \\ - L_{0} (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \\ - [A_{0} x (t) + B_{0} u (t) + B_{1} d_{1} (t) + B_{2} d_{2} (t)] \\ = [A_{0} - L_{0} C_{0}] \tilde{x} (t) + [B_{1} V_{1} - L_{0} C_{1} V_{1}] {\tilde{ω}}_{1} (t) \\ + [B_{2} V_{2} - L_{0} C_{2} V_{2}] {\tilde{ω}}_{2} (t) \\ {\overset{\cdot}{\tilde{ω}}}_{1} (t) = G_{1} {\hat{ω}}_{1} (t) - L_{1} (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \\ - G_{1} ω_{1} - H_{1} δ_{1} (t) \\ = - L_{1} C_{0} \tilde{x} (t) + [G_{1} - L_{1} C_{1} V_{1}] {\tilde{ω}}_{1} (t) \\ - L_{1} C_{2} V_{2} {\tilde{ω}}_{2} (t) - H_{1} δ_{1} (t) \\ {\overset{\cdot}{\tilde{ω}}}_{2} (t) = G_{2} {\hat{ω}}_{2} (t) - L_{2} (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \\ - G_{2} ω_{2} (t) - f (ω_{2} (t), x (t), u (t)) - H_{2} δ_{2} (t) \\ - β (t) (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \\ φ ({\hat{ω}}_{2} (t), y (t), u (t)) \\ = - L_{2} C_{0} \tilde{x} (t) - L_{2} C_{1} V_{1} {\tilde{ω}}_{1} (t) + [G_{2} - L_{2} C_{2} V_{2}] {\tilde{ω}}_{2} (t) \\ - f (ω_{2} (t), x (t), u (t)) - H_{2} δ_{2} (t) \\ - β (t) (C_{0} \hat{x} (t) + C_{1} V_{1} {\hat{ω}}_{1} (t) + C_{2} V_{2} {\hat{ω}}_{2} (t) - y (t)) \\ φ ({\hat{ω}}_{2} (t), y (t), u (t)) \end{matrix}

(9)

Combining (1) and (8) yields

\begin{matrix} \overset{\cdot}{x} (t) = (A_{0} + B_{0} K_{0}) x (t) + B_{0} K_{0} \tilde{x} (t) + (B_{1} - B_{0} K_{1}) d_{1} (t) \\ - B_{0} K_{1} V_{1} {\tilde{ω}}_{1} (t) + (B_{2} - B_{0} K_{2}) d_{2} (t) - B_{0} K_{2} V_{2} {\tilde{ω}}_{2} (t) \end{matrix}

(10)

Considering that $rank (B_{0}, B_{1}) = rank (B_{0}, B_{2}) = rank (B_{0}) = m$ , we can select the gain $K_{1}, K_{2}$ such that $B_{1} - B_{0} K_{1} = 0$ and $B_{2} - B_{0} K_{2} = 0$ . Therefore, equation (10) turns to

\begin{matrix} \overset{\cdot}{x} (t) = (A_{0} + B_{0} K_{0}) x (t) + B_{0} K_{0} \tilde{x} (t) \\ - B_{0} K_{1} V_{1} {\tilde{ω}}_{1} (t) - B_{0} K_{2} V_{2} {\tilde{ω}}_{2} (t) \end{matrix}

(11)

By defining $ξ (t) = {[x^{T} (t), {\tilde{x}}^{T} (t), {\tilde{ω}}_{1}^{T} (t), {\tilde{ω}}_{2}^{T} (t)]}^{T}$ , we can get the following composite system

\begin{matrix} \overset{\cdot}{ξ} (t) = A ξ (t) - Bf (ω_{2} (t), y (t), u (t)) \\ - β (t) BC ξ (t) φ ({\hat{ω}}_{2} (t), y (t), u (t)) - H δ (t) \end{matrix}

(12)

where

\begin{matrix} A = [\begin{matrix} A_{0} + B_{0} K_{0} & B_{0} K_{0} & - B_{0} K_{1} V & - B_{0} K_{2} V_{2} \\ 0 & A_{0} - L_{0} C_{0} & B_{1} V_{1} - L_{0} C_{1} V_{1} & B_{2} V_{2} - L_{0} C_{2} V_{2} \\ 0 & - L_{1} C_{0} & G_{1} - L_{1} C_{1} V_{1} & - L_{1} C_{2} V_{2} \\ 0 & - L_{2} C_{0} & - L_{2} C_{1} V_{1} & G_{2} - L_{2} C_{2} V_{2} \end{matrix}], \\ 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}], \\ C = [0^{n \times n}, C_{0}, C_{1}, C_{2}] \end{matrix}

(13)

Theorem 1: Given nonlinear system (1) subject to non-harmonic multisource disturbances generated by (2) and (3). If there exist positive-definite matrices $Q, P$ such that the following equalities hold

\begin{matrix} A^{T} P + PA = - Q \\ PB = C^{T} \end{matrix}

(14)

then the proposed output feedback anti-disturbance control laws (5)–(8) can guarantee that all the signals in the closed-loop system remain bounded.

Proof: Define a Lyapunov function candidate as

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

(15)

where $β^{*} \geq max {c_{2} ρ_{1}, c_{4} ρ_{2}} / 2$ . $ρ_{1}$ and $ρ_{2}$ are positive constants. Hence, along (12), we can take the derivative of V as

\begin{matrix} \overset{\cdot}{V} = - \frac{1}{2} ξ^{T} (t) Q ξ (t) - ξ {(t)}^{T} PBf (ω_{2} (t), y (t), u (t)) \\ - β (t) ξ^{T} (t) PBC ξ (t) φ ({\hat{ω}}_{2} (t), y (t), u (t)) \\ - ξ^{T} (t) PBH δ (t) + η^{- 1} (β (t) - β^{*}) \overset{\cdot}{β} (t) \end{matrix}

(16)

By using Assumption 2, we can get the following inequality from (14)

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{2} ξ^{T} (t) Q ξ (t) + ‖ C ξ (t) ‖ [c_{1} + c_{2} ‖ ω_{2} (t) ‖ \\ + c_{3} g_{1} (y (t), u (t)) + c_{4} ‖ ω_{2} (t) ‖ g_{2} (y (t), u (t))] \\ - β (t) {[C ξ (t)]}^{T} [C ξ (t)] φ ({\hat{ω}}_{2} (t), y (t), u (t)) \\ - ξ^{T} (t) PBH δ (t) + η^{- 1} (β (t) - β^{*}) \overset{\cdot}{β} (t) \end{matrix}

(17)

Then with the aid of $‖ ω_{2} (t) ‖ \leq ‖ {\hat{ω}}_{2} (t) ‖ + ‖ {\tilde{ω}}_{2} (t) ‖$ , we can get that

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{2} ξ^{T} (t) Q ξ (t) + ‖ C ξ (t) ‖ \\ [\begin{matrix} c_{1} + c_{2} ‖ {\hat{ω}}_{2} (t) ‖ + c_{2} ‖ {\tilde{ω}}_{2} (t) ‖ + c_{3} g_{1} (y (t), u (t)) \\ + c_{4} ‖ {\hat{ω}}_{2} (t) ‖ g_{2} (y (t), u (t)) + c_{4} ‖ {\tilde{ω}}_{2} (t) ‖ g_{2} (y (t), u (t)) \end{matrix}] \\ - β (t) {[C ξ (t)]}^{T} [C ξ (t)] φ ({\hat{ω}}_{2} (t), y (t), u (t)) \\ - ξ^{T} (t) PBH δ (t) + η^{- 1} (β (t) - β^{*}) \overset{\cdot}{β} (t) \end{matrix}

(18)

Based on Young’s inequalities, it can be checked that for any $ρ_{1}, ρ_{2}, ρ_{3} > 0$

\begin{matrix} c_{2} ‖ C ξ (t) ‖ ‖ {\tilde{ω}}_{2} (t) ‖ \leq \frac{1}{2} c_{2} (ρ_{1} {‖ C ξ (t) ‖}^{2} + \frac{1}{ρ_{1}} {‖ {\tilde{ω}}_{2} (t) ‖}^{2}) \\ c_{4} ‖ C ξ (t) ‖ ‖ {\tilde{ω}}_{2} (t) ‖ g_{2} (y (t), u (t)) \\ \leq \frac{1}{2} c_{4} (ρ_{2} {‖ C ξ (t) ‖}^{2} g_{2}^{2} (y (t), u (t)) + \frac{1}{ρ_{2}} {‖ {\tilde{ω}}_{2} (t) ‖}^{2}) \\ ‖ ξ^{T} (t) PBH δ (t) ‖ \leq \frac{1}{2} (ρ_{3} {\bar{δ}}^{2} + \frac{ξ^{T} (t) ξ (t)}{ρ_{3}}) \end{matrix}

where $\bar{δ} =$ $sup_{t \geq 0} ‖ PBH δ (t) ‖$ . Without losing generality, we can select $ρ_{1}, ρ_{2}$ and $ρ_{3}$ such that $c_{2} / ρ_{1} + c_{4} / ρ_{2} \leq λ_{min} (Q) / 4, 1 / ρ_{3} \leq λ_{min} (Q) / 4$ . Therefore, it can be proven that

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) + ‖ C ξ (t) ‖ [c_{1} + c_{2} ‖ {\hat{ω}}_{2} (t) ‖ \\ + c_{3} g_{1} (y (t), u (t)) + c_{4} ‖ {\hat{ω}}_{2} (t) ‖ g_{2} (y (t), u (t))] \\ - β (t) {[C ξ (t)]}^{T} [C ξ (t)] φ ({\hat{ω}}_{2} (t), y (t), u (t)) \\ + η^{- 1} (β (t) - β^{*}) \overset{\cdot}{β} (t) + \frac{1}{2} c_{2} ρ_{1} ‖ C ξ (t) ‖^{2} \\ + \frac{1}{2} c_{4} ρ_{2} ‖ C ξ (t) ‖^{2} g_{2}^{2} (y (t), u (t)) + \frac{1}{2} ρ_{3} {\bar{δ}}^{2} \end{matrix}

(20)

Combining (6), (7) and (20) yields

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) + ‖ C ξ (t) ‖ [c_{1} + c_{2} ‖ {\hat{ω}}_{2} (t) ‖ \\ + c_{3} g_{1} (y (t), u (t)) + c_{4} ‖ {\hat{ω}}_{2} (t) ‖ g_{2} (y (t), u (t))] \\ - β^{*} ‖ C ξ (t) ‖^{2} [2 + {‖ {\hat{ω}}_{2} (t) ‖}^{2} + g_{1}^{2} (y (t), u (t)) \\ + g_{2}^{2} (y (t), u (t)) + {‖ {\hat{ω}}_{2} (t) ‖}^{2} g_{2}^{2} (y (t), u (t))] \\ - σ (β (t) - β^{*}) β (t) + \frac{1}{2} c_{2} ρ_{1} ‖ C ξ (t) ‖^{2} \\ + \frac{1}{2} c_{4} ρ_{2} ‖ C ξ (t) ‖^{2} g_{2}^{2} (y (t), u (t)) + \frac{1}{2} ρ_{3} {\bar{δ}}^{2} \end{matrix}

(21)

Obviously

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) - (β^{*} - \frac{1}{2} c_{2} ρ_{1}) ‖ C ξ (t) ‖^{2} \\ - (β^{*} - \frac{1}{2} c_{4} ρ_{2}) ‖ C ξ (t) ‖^{2} g_{2}^{2} (y (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}

(22)

Considering that $β^{*} \geq max {c_{2} ρ_{1}, c_{4} ρ_{2}} / 2$ and $- 2 (β (t) - β^{*}) β (t) \leq - {(β (t) - β^{*})}^{2} + {(β^{*})}^{2}$ , we can obtain the following inequality

\overset{\cdot}{V} \leq - μ V + M

(23)

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}

Solving (23) implies that

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

(24)

Clearly, $0 \leq V (t) \leq max {M / μ, V (0)}$ for all $t \geq 0$ and the estimation errors and the system states are all bounded. The proof is complete.

Remark 1: Note that the proposed control approach is in fact an output controller. It is supposed that the system states are not available, and an augmented state observer is constructed to estimate both of the systems states and the disturbances from the system outputs. In the state-feedback controller, the estimates of the systems states are used, rather than the systems states themselves.

Patulous results for a class of intense non-harmonic disturbances

In practical engineering systems, the dynamics of the non-harmonic disturbances are often influenced by multiple variables, including the inputs, outputs and the system states. Unfortunately, the results obtained in Section 3 are based on the assumption that the non-harmonic disturbances are merely influenced by the outputs and inputs of the system. To over this limitation, aiming at a class of intense non-harmonic disturbances, which are influenced by the inputs, outputs and the system states simultaneously, a patulous robust adaptive anti-disturbance control structure is constructed in this section.

Before designing the controller, we make the following assumption

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

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

(25)

where $f (ω_{2} (t), x (t), u (t)) \in ℜ^{p}$ is an unknown nonlinear function. $G_{2}, V_{2}, H_{2}, δ_{2} (t)$ are defined as the same as (3). It is supposed that

\begin{matrix} ‖ f (ω_{2} (t), x (t), u (t)) ‖ \leq c_{1} + c_{2} ‖ ω_{2} (t) ‖ + c_{3} ‖ x (t) ‖ \\ + c_{4} g_{1} (y (t), u (t)) + c_{5} ‖ ω_{2} ‖ g_{2} (y (t), u (t)) \\ + c_{6} ‖ x ‖ g_{3} (y (t), u (t)) \end{matrix}

(26)

where $g_{i} (y (t), u (t)), 1 \leq i \leq 3$ are known nonnegative functions, $c_{i}, 1 \leq i \leq 6$ are unknown constants.

Aiming at the nonlinear system (1) under Assumptions 1 and 3, the anti-disturbance control law, the parameter update law, and the adaptive nonlinear augmented state observer are designed as the same as before. A novel $φ (\cdot)$ is developed as

\begin{matrix} φ ({\hat{ω}}_{2} (t), \hat{x} (t), y (t), u (t)) = 2 + ‖ {\hat{ω}}_{2} (t) ‖^{2} + ‖ \hat{x} (t) ‖^{2} \\ + g_{1}^{2} (y (t), u (t)) + g_{2}^{2} (y (t), u (t)) \\ + ‖ {\hat{ω}}_{2} (t) ‖^{2} g_{2}^{2} (y (t), u (t)) + g_{3}^{2} (y (t), u (t)) \\ + ‖ \hat{x} (t) ‖^{2} g_{3}^{2} (y (t), u (t)) \end{matrix}

(27)

Then, with the aid of similar arguments as Subsection 3, the following closed-loop system can be obtained

\begin{matrix} \overset{\cdot}{ξ} (t) = A ξ (t) - Bf (ω_{2} (t), x (t), u (t)) \\ - β (t) BC ξ (t) φ ({\hat{ω}}_{2} (t), \hat{x} (t), y (t), u (t)) - H δ (t) \end{matrix}

(28)

Theorem 2: Consider nonlinear system (1) subjected to multiple disturbance generated from (2) and (3). If there exist positive-definite matrices $Q, P$ satisfying (14), then the proposed output feedback anti-disturbance control law (5), (7), (8) and (27) can guarantee that all the signals in the closed-loop system remain bounded.

Proof: Let us select the following Lyapunov function

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

(29)

where $β^{*} \geq max {c_{2} ρ_{1} + c_{3} ρ_{2}, c_{5} ρ_{3}, c_{6} ρ_{4}} / 2$ . $ρ_{i}, 1 \leq i \leq 4$ are positive constants satisfying $c_{2} / ρ_{1} + c_{3} / ρ_{2} + c_{5} / ρ_{3} + c_{6} / ρ_{4} \leq λ_{min} (Q) / 4$ and $1 / ρ_{5} \leq λ_{min} (Q) / 4$ . Accordingly, we can obtain the following equality

\begin{matrix} \overset{\cdot}{V} = - \frac{1}{2} ξ^{T} (t) Q ξ (t) - ξ {(t)}^{T} PBf (ω_{2} (t), x (t), u (t)) \\ - ξ^{T} (t) PBH δ (t) + η^{- 1} (β (t) - β^{*}) \overset{\cdot}{β} (t) \\ - β (t) ξ^{T} (t) PBC ξ (t) φ ({\hat{ω}}_{2} (t), \hat{x} (t), y (t), u (t)) \end{matrix}

(30)

Then, based on Assumption 3, we know that

\begin{matrix} \overset{\cdot}{V} ⩽ - \frac{1}{2} ξ^{T} (t) Q ξ (t) + ‖ C ξ (t) ‖ [c_{1} + c_{2} ‖ ω_{2} ‖ + c_{3} ‖ x ‖ \\ + c_{4} g_{1} (y, u) + c_{5} ‖ ω_{2} ‖ g_{2} (y, u) + c_{6} ‖ x ‖ g_{3} (y, u)] \\ - β (t) {[C ξ (t)]}^{T} [C ξ (t)] φ ({\hat{ω}}_{2} (t), \hat{x} (t), y (t), u (t)) \\ - ξ^{T} (t) PBH δ (t) + η^{- 1} (β (t) - β^{*}) \overset{\cdot}{β} (t) \end{matrix}

(31)

Considering that $‖ ω_{2} (t) ‖ \leq ‖ {\hat{ω}}_{2} (t) ‖ + ‖ {\tilde{ω}}_{2} (t) ‖$ and $‖ x (t) ‖ \leq ‖ \hat{x} (t) ‖ + ‖ \tilde{x} (t) ‖$ , we can get the following inequality

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{2} ξ^{T} (t) Q ξ (t) + ‖ C ξ (t) ‖ \\ [\begin{matrix} c_{1} + c_{2} ‖ {\hat{ω}}_{2} ‖ + c_{2} ‖ {\tilde{ω}}_{2} ‖ + c_{3} ‖ \hat{x} ‖ + c_{3} ‖ \tilde{x} ‖ + c_{4} g_{1} (y, u) \\ + c_{5} ‖ {\hat{ω}}_{2} ‖ g_{2} (y, u) + c_{5} ‖ {\tilde{ω}}_{2} ‖ g_{2} (y, u) + c_{6} ‖ \hat{x} ‖ g_{3} (y, u) \\ + c_{6} ‖ \tilde{x} ‖ g_{3} (y, u) \end{matrix}] \\ - β (t) {[C ξ (t)]}^{T} [C ξ (t)] φ ({\hat{ω}}_{2} (t), \hat{x} (t), y (t), u (t)) \\ - ξ^{T} (t) PBH δ (t) + η^{- 1} (β (t) - β^{*}) \overset{\cdot}{β} (t) \end{matrix}

(32)

With the aid of the Young’s inequalities, it can be proven that for any $ρ_{i}, 1 \leq i \leq 5$

\begin{matrix} c_{2} ‖ C ξ (t) ‖ ‖ {\tilde{ω}}_{2} (t) ‖ \leq \frac{1}{2} c_{2} (ρ_{1} {‖ C ξ (t) ‖}^{2} + \frac{1}{ρ_{1}} {‖ {\tilde{ω}}_{2} (t) ‖}^{2}) \\ c_{3} ‖ C ξ (t) ‖ ‖ \tilde{x} (t) ‖ \leq \frac{1}{2} c_{3} (ρ_{2} {‖ C ξ (t) ‖}^{2} + \frac{1}{ρ_{2}} {‖ \tilde{x} (t) ‖}^{2}) \\ c_{5} ‖ C ξ (t) ‖ ‖ {\tilde{ω}}_{2} (t) ‖ g_{2} (y, u) \\ \leq \frac{1}{2} c_{5} (ρ_{3} {‖ C ξ (t) ‖}^{2} g_{2}^{2} (y, u) + \frac{1}{ρ_{3}} {‖ {\tilde{ω}}_{2} (t) ‖}^{2}) \\ c_{6} ‖ C ξ (t) ‖ ‖ \tilde{x} (t) ‖ g_{3} (y, u) \\ \leq \frac{1}{2} c_{6} (ρ_{4} {‖ C ξ (t) ‖}^{2} g_{3}^{2} (y, u) + \frac{1}{ρ_{4}} {‖ \tilde{x} (t) ‖}^{2}) \\ ‖ ξ^{T} (t) PBH δ (t) ‖ \leq \frac{1}{2} (ρ_{5} {\bar{δ}}^{2} + \frac{ξ^{T} (t) ξ (t)}{ρ_{5}}) \end{matrix}

(33)

where $\bar{δ} =$ $sup_{t \geq 0} ‖ PBH δ (t) ‖$ . Considering the fact that $c_{2} / ρ_{1} + c_{3} / ρ_{2} + c_{5} / ρ_{3} + c_{6} / ρ_{4} \leq λ_{min} (Q) / 4$ and $1 / ρ_{5} \leq λ_{min} (Q) / 4$ , it is easy to get that

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) + ‖ C ξ (t) ‖ [c_{1} + c_{2} ‖ {\hat{ω}}_{2} ‖ + c_{3} ‖ \hat{x} ‖ \\ + c_{4} g_{1} (y, u) + c_{5} ‖ {\hat{ω}}_{2} ‖ g_{2} (y, u) + c_{6} ‖ \hat{x} ‖ g_{3} (y, u)] \\ - β (t) {[C ξ (t)]}^{T} [C ξ (t)] φ ({\hat{ω}}_{2} (t), \hat{x} (t), y (t), u (t)) \\ + η^{- 1} (β (t) - β^{*}) \overset{\cdot}{β} (t) + \frac{1}{2} (c_{2} ρ_{1} + c_{3} ρ_{2}) ‖ C ξ (t) ‖^{2} \\ + \frac{1}{2} c_{5} ρ_{3} ‖ C ξ (t) ‖^{2} g_{2}^{2} (y, u) + \frac{1}{2} c_{6} ρ_{4} ‖ C ξ (t) ‖^{2} g_{3}^{2} (y, u) + \frac{1}{2} ρ_{5} {\bar{δ}}^{2} \end{matrix}

(34)

Then from (27) and (7), we know that

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) + ‖ C ξ (t) ‖ [c_{1} + c_{2} ‖ {\hat{ω}}_{2} ‖ + c_{3} ‖ \hat{x} ‖ + c_{4} g_{1} (y, u) + c_{5} ‖ {\hat{ω}}_{2} ‖ g_{2} (y, u) + c_{6} ‖ \hat{x} ‖ g_{3} (y, u)] \\ - β^{*} ‖ C ξ (t) ‖^{2} [\begin{matrix} 2 + ‖ {\hat{ω}}_{2} (t) ‖^{2} + ‖ \hat{x} (t) ‖^{2} + g_{1}^{2} (y (t), u (t)) + g_{2}^{2} (y (t), u (t)) \\ + ‖ {\hat{ω}}_{2} (t) ‖^{2} g_{2}^{2} (y (t), u (t)) + g_{3}^{2} (y (t), u (t)) + ‖ \hat{x} (t) ‖^{2} g_{3}^{2} (y (t), u (t)) \end{matrix}] \\ + \frac{1}{2} (c_{2} ρ_{1} + c_{3} ρ_{2}) ‖ C ξ (t) ‖^{2} + \frac{1}{2} c_{5} ρ_{3} ‖ C ξ (t) ‖^{2} g_{2}^{2} (y, u) + \frac{1}{2} c_{6} ρ_{4} ‖ C ξ (t) ‖^{2} g_{3}^{2} (y, u) + \frac{1}{2} ρ_{5} {\bar{δ}}^{2} \\ - σ (β (t) - β^{*}) β (t) \end{matrix}

(35)

Further derivations imply that

\begin{matrix} \overset{\cdot}{V} \leq - \frac{1}{4} ξ^{T} (t) Q ξ (t) - (β^{*} - \frac{1}{2} c_{2} ρ_{1} - \frac{1}{2} c_{3} ρ_{2}) ‖ C ξ (t) ‖^{2} \\ - (β^{*} - \frac{1}{2} c_{5} ρ_{3}) ‖ C ξ (t) ‖^{2} g_{2}^{2} (y, u) \\ - (β^{*} - \frac{1}{2} c_{6} ρ_{4}) ‖ C ξ (t) ‖^{2} g_{3}^{2} (y, u) + \frac{1}{4 β^{*}} \sum_{i = 1}^{6} c_{i}^{2} \\ + \frac{1}{2} ρ_{5} {\bar{δ}}^{2} - σ (β (t) - β^{*}) β (t) \end{matrix}

(36)

By using $β^{*} \geq max {c_{2} ρ_{1} + c_{3} ρ_{2}, c_{5} ρ_{3}, c_{6} ρ_{4}} / 2$ and $- 2 (β (t) - β^{*}) β (t) \leq - {(β (t) - β^{*})}^{2} + {(β^{*})}^{2}$ , it can be checked that

\overset{\cdot}{V} \leq - μ V + M

(37)

where

\begin{matrix} μ = min {\frac{λ_{min} (Q)}{2 λ_{max} (P)}, σ η} \\ M = \frac{1}{4 β^{*}} \sum_{i = 1}^{6} c_{i}^{2} + \frac{1}{2} ρ_{5} {\bar{δ}}^{2} + \frac{1}{2} σ {[β^{*}]}^{2} \end{matrix}

Hence, by using the similar arguments as Theorem 1, it can be concluded that the estimation errors and the system states are all bounded. The proof is complete.

Remark 2: In the output feedback case, since the system states are not available, they cannot be applied directly in the nonlinear function (27). As a result, more bound parameters have to be used and the proof turns to be more complex. This point can be observed easily in the proof of Theorem 2. Therefore, it can be concluded that when the state $x (t)$ was included the nonlinear features may be more intense than $y (t)$ .

Simulations

To illustrative the effectiveness of the proposed approach, a practical example is employed in this section. In this paper, we consider a simplified longitudinal dynamic equation of a back-to-turn missile, which can be formulated by

\begin{matrix} \overset{\cdot}{α} = ω_{z} \\ {\overset{\cdot}{ω}}_{z} = a_{1} α + a_{2} ω_{z} + b_{1} u + d_{1} + d_{2} \end{matrix}

(38)

where $α$ is angle of attack, $ω_{z}$ is the angular velocity around z axis. u is the output of the rudder system. $a_{1}, a_{2}, b_{1}$ are the dynamic parameters. It is assumed that the non-harmonic multiple disturbances $d_{1}$ and $d_{2}$ exist in this longitudinal model. $y_{1} = α + 0.1 d_{1}, y_{2} = ω_{z} + 0.05 d_{2}$ . The coefficient matrices of harmonic disturbances $d_{1} (t)$ are supposed to be $G_{1} = [0 0.7; - 0.7 0], V_{1} = [2 2], H_{1} = {[0.005 0.005]}^{T}$ . In the simulation, $δ_{1} (t), δ_{2} (t)$ as selected as sinusoidal disturbances. The initial values of the system states are set as $α (0) = 0.2, ω_{z} = 0.05$ . The dynamic parameters are taken as $a_{1} = 0.85, a_{2} = 0.48, b_{1} = 0.76$ .

In Case 1, the usual non-harmonic disturbances are considered. The dynamic equations of the non-harmonic disturbance $d_{2} (t)$ are given as

\begin{matrix} {\overset{\cdot}{ω}}_{2, 1} = 0.2 ω_{2, 1} + 0.3 ω_{2, 2} \\ {\overset{\cdot}{ω}}_{2, 2} = 0.3 ω_{2, 2} (ω_{2, 1} + ω_{2, 2}) - 3 ω_{2, 1} ω_{2, 2} \\ + 0.05 (ω_{2, 1} + ω_{2, 2}) (y_{1} + y_{2} + \cos (u^{2})) + 0.005 \sin (t) \end{matrix}

(39)

From the results given in Section 3, we select that $K = [- 27.4342, - 22.9474], K_{1} = 1, K_{2} = 1$ . The nonlinear damping term is developed as

\begin{matrix} φ ({\hat{ω}}_{2} (t), x (t), u (t)) = 2 + ‖ {\hat{ω}}_{2} (t) ‖^{2} \\ + 0.0025 {(ω_{2, 1} + ω_{2, 2})}^{2} {(y_{1} + y_{2} + \cos (u^{2}))}^{2} \\ + 0.0004 ω_{2, 2}^{2} {(ω_{2, 1} + ω_{2, 2})}^{2} \end{matrix}

(40)

Moreover, we choose $η = 3, σ = 8$ . The simulation results are provided in Figure 1 –Figure 3.

In Case 2, the intense non-harmonic disturbances are taken into account. It is supposed that $d_{2} (t)$ possess the following dynamic equations

\begin{matrix} {\overset{\cdot}{ω}}_{2, 1} = {\overset{\cdot}{ω}}_{2, 2} \\ {\overset{\cdot}{ω}}_{2, 2} = \frac{- 2 (ω_{2, 1} + ω_{2, 2})}{2 - e^{- 5 t}} \\ + 0.5 (ω_{2, 1} + ω_{2, 2}) \cos (y_{1} + y_{2} + \sin (0.1 u^{2})) \\ + 0.8 (y_{1}^{2} + y_{2}^{2}) \cos (u) \end{matrix}

(41)

The longitudinal dynamic equation (38) is still under consideration. The gain matrices are selected as same as Case 1. The initial conditions of the system remain. Figure 4 –Figure 6 demonstrate the control performance and the disturbance estimation results.

From the simulation results, it is easy to find that the system states and the disturbance estimation errors stay bounded and can converge to a small value, even if influenced by the harmonic and non-harmonic disturbances simultaneously. Furthermore, by considering different kinds if non-harmonic disturbances, the robustness of the proposed method can be verified.

Conclusion

This paper has studied the problem of adaptive output-feedback control for a class of uncertain nonlinear systems. The nonlinear systems are supposed to suffer from harmonic and non-harmonic multisource disturbances simultaneously. A new adaptive anti-disturbance control technique based on the nonlinear damping term and an augmented state observer was established to guarantee that all the signals in the resulting closed-loop system remain bounded. It is worth noting that the results have been extended to a class of intense nonlinear non-harmonic disturbances. Finally, two numerical examples have been presented to demonstrate the effectiveness of the proposed technique. It is noted that the problem of finite-time control for nonlinear systems subjected to multisource disturbances is an untreated topic, which will be our future research work.

Footnotes

Declaration of conflicting interest

The author declares that there is no conflict of interest.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China under Grants no. 61473226 and no 61503302.

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