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Composite stratified anti‐disturbance control for a class of MIMO discrete‐time systems with nonlinearity 标题和描述提到的研究论文内容涉及了多输入多输出（MIMO）离散时间系统在存在非线性情况下的复合分层抗干扰控制方法。在这篇论文中，研究者们探讨了如何对受到多源干扰的非线性系统进行抗干扰控制和估计问题。文中提出了一个基于扰动观测器的控制方案，这种方案能够估计出具有部分已知信息的扰动，并且将离散时间滑模控制（DSMC）与基于扰动观测器的控制结合起来，形成了一种新颖的控制策略。从这些信息中，可以总结出以下几点重要的知识点： 1. 非线性系统控制和抗干扰研究论文的焦点是非线性系统的抗干扰控制问题。非线性系统在实际应用中，如机器人、飞行控制系统、工业过程控制中非常常见，且往往受到外部和内部多种干扰源的影响，这使得系统控制复杂且困难。因此，如何有效抑制和消除干扰，确保系统稳定和精准运行，是控制领域研究的热点问题。 2. 扰动观测器（Disturbance Observer）扰动观测器是一种用于估计和补偿系统扰动的控制策略。在控制工程中，扰动观测器基于系统模型和反馈信息，对系统的未知扰动进行观测和估计，然后通过控制输入对估计到的扰动进行补偿，以减少扰动对系统性能的影响。在论文中，扰动观测器是独立于控制器设计构建的，这意味着它在控制器设计之外，但可以利用部分已知信息估计出干扰。 3. 离散时间滑模控制（DSMC）离散时间滑模控制是一种有效的非线性控制方法，通常在数字控制系统中使用。滑模控制的核心在于设计一个切换函数，使得系统状态在有限时间内达到并保持在切换面的滑动模式上。滑模控制的优点在于对参数变化和外部干扰具有很强的鲁棒性。论文将离散时间滑模控制与扰动观测器结合起来，用于构建复合分层的抗干扰控制策略。 4. 复合分层抗干扰控制策略复合分层抗干扰控制策略是一种综合了多种控制方法的先进控制策略。该策略不仅考虑了已知的非线性动态，还考虑了未知的非线性动态，可以适应更复杂的系统环境。论文提出的控制策略通过整合不同的控制技术，对系统提供了更全面的保护。 5. MIMO系统多输入多输出（MIMO）系统指的是有多个输入信号和多个输出信号的系统。与单输入单输出（SISO）系统相比，MIMO系统能提供更高的数据传输速率和更好的性能。在论文中，提出的控制策略正是针对MIMO系统设计的，这对于提高系统的控制性能有重要意义。 6. 仿真与飞行控制系统论文通过飞行控制系统的仿真实例，验证了所提出控制方案的有效性。飞行控制系统是一个典型的非线性、多变量、多扰动的复杂控制系统。仿真实验通过与以往的控制方案比较，突显了新提出的控制策略在性能上的优势。 7. 关键词分析论文中的关键词“compositestratiﬁedcontrol”、“disturbance-observer-basedcontrol”、“discrete-timesliding-modecontrol”和“MIMOsystemswithnonlinearity”，反映了论文研究的核心内容和重点。这些关键词为我们提供了理解论文主要研究方向和技术要点的线索。综合以上知识点，研究者们针对具有非线性动态和受到多源干扰的MIMO离散时间系统，提出了一种新型的复合分层抗干扰控制策略。该策略通过整合扰动观测器和离散时间滑模控制，为解决非线性系统在干扰条件下的控制问题提供了一种有效的方法，并通过飞行控制系统的仿真实例证明了其有效性。这些研究不仅对于飞行控制系统有着重要应用价值，而且对于其他各类非线性控制系统的性能提升也具有普遍的借鉴意义。

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Composite stratiﬁed anti-disturbance control for a class of MIMO

discrete-time systems with nonlinearity

Xinjiang Wei

1,2,∗, †

, Nan Chen

, Changhui Deng

, Xiaohua Liu

and Meiqin Tang

Department of Mathematics and Information, Ludong University, Yantai 264025, People’s Republic of China

School of Mechanical Engineering, Southeast University, Nanjing 210096, People’s Republic of China

School of Information Engineering, Dalian Ocean University, Dalian 116023, People’s Republic of China

SUMMARY

Anti-disturbance control and estimation problem are investigated for nonlinear system subject to multi-

source disturbances. The disturbances classiﬁed model is proposed based on the error and noise analysis

of priori knowledge. The disturbance observers are constructed separately from the controller design to

estimate the disturbance with partial known information. By integrating disturbance-observer-based control

with discrete-time sliding-mode control (DSMC), a novel type of composite stratiﬁed anti-disturbance

control scheme is presented for a class of multiple-input–multiple-output discrete-time systems with

known and unknown nonlinear dynamics, respectively. Simulations for a ﬂight control system are given

to demonstrate the effectiv eness of the results compared with the previous schemes. Copyright 䉷 2011

John Wiley & Sons, Ltd.

Receiv e d 6 October 2009; Revised 9 December 2010; Accepted 5 January 2011

KEY WORDS

: composite stratiﬁed control; disturbance-observer-based control; discrete-time sliding-

mode control; MIMO systems with nonlinearity

1. INTRODUCTION

Control of nonlinear systems under disturbance has been an active research area in the last decades

and several elegant approaches have been presented [1–5]. Basically, control methods addressing

disturbance attenuation can be classiﬁed according to the assumptions made on disturbances.

The ﬁrst approach is to model disturbances by a stochastic process, which leads to nonlinear

stochastic control theory [1]. The second approach is nonlinear H

∞

control, which can attenuate

the inﬂuences from disturbances to controlled output at a desired level for systems with bounded

disturbances [3, 4]. The third approach is constructive nonlinear control theory based on control

Lyapunov function concept, where a nonlinear controller can be designed by a backstepping

procedure [5]. Many of which have been extensively investigated and many signiﬁcant results have

been developed. However, in general, design of controllers using these approaches involves the

solution of a set of nonlinear partial differential equations (PDEs) or nonlinear partial differential

inequalities (PDIs). Although signiﬁcant progress has been achieved in this ﬁeld, solution of those

PDEs or PDIs, in general, can only be obtained for special cases or approximation. In addition,

only stability of the nominal system in the absence of the disturbances can be guaranteed by the

aforementioned approaches. Hence, the improved anti-disturbance performance is desired for the

high-accuracy control.

∗

Correspondence to: Xinjiang Wei, Department of Mathematics and Information, Ludong University, Yantai 264025,

People’s Republic of China.

†

E-mail: weixinjiang@163.com

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL

Published online 18 February 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1709

Int. J. Robust. Nonlinear Control

2012; :

–

22 453 472

X. WEI ET AL.

Disturbance observer was established in the late of 1980s for linear frequency-domain systems,

has a simple structure and is easily implemented in engineering [6]. The motivation is suggested by

the fact that if the disturbances can be estimated, then the control of the uncertain dynamic systems

with disturbances may become easier [7]. In the past few years, disturbance-observer-based control

(DOBC) strategies were successfully used in achieving robust stability and performance in motion

control systems, for instance, robot manipulators [8], CNC machine centers in cutting process

[9], high-speed direct-drive positioning table [10], permanent magnet synchronous motor [11],

magnetic hard drive servo system [12] etc. Reference [13] gives a uniﬁed and historical review of

disturbance observer design for the beneﬁt of practitioners. It appears that disturbance observers

mentioned above are mostly designed for linear systems. Recently, DOBC has been extended from

linear systems to nonlinear systems [14, 15]. In [14], single-input–single-output nonlinear systems

with well-deﬁned disturbance relative degree were studied, and the disturbances were limited to be

constant or harmonic signals. In [15], the DOBC approaches for a class of multiple-input–multiple-

output (MIMO) nonlinear systems have been considered and the disturbances were represented by

a linear exogenous system. This extended the assumptions of the disturbances, which were limited

to be constant, harmonic in [16, 17]. However, it has been reported that when the disturbance has

perturbations, the proposed approaches in DOBC [14, 15] are unsatisfactory.

Sliding-mode control (SMC) is well known for their robustness to system parameter variations

and disturbances, which helps in achieving a satisfactory level of robustness and invariant behavior

by a simple method of changing the structure of the control according to the system states [18].

Based o n the robustness of SMC, a type of control scheme combining the DOBC with terminal

SMC is proposed for a class of MIMO continuous nonlinear systems in [19]. Besides continuous-

time SMC, research interest has developed for the area of discrete-time sliding-mode control

(DSMC) in the recent years [20–22]. The invariance property presented in continuous time sliding

mode is not preserved in discrete-time sliding m ode. Here, the system states move about the sliding

manifold are unable to stay on it; hence, giving the terminology quasi-sliding mode (QSM) [23].

Few researchers have also attempted the design of SMC of discrete-time systems with unmatched

uncertainties [24].

The disturbances concerning above are m ostly from single source, so can be estimated and

attenuated by a single controller. However, multi-source disturbances widely exist in circuits and

systems, telecommunications, signal processing, the control community, etc. Usually these distur-

bances are classiﬁed into two categories based on the error and noise analysis of future knowledge.

One are often assumed to generated by an exogenous system with uncertainty, which can represent

the disturbances with partial known information. The other is supposed to be the bounded H

norm

or random model, which can represent the disturbances with unknown information. The two type

of controller can be designed for the two type of disturbances model, then integrated as a composite

controller. However, the coupling from different disturbance controller can cause complex structure

of closed-loop system [25]. The stability and the disturbance attenuation performance guaranteed

by single controller cannot be guaranteed by the composite controller.

Considering most of the aforementioned research results about DOBC is concerned with contin-

uous systems, in this paper, a novel type of composite stratiﬁed anti-disturbance control scheme

is proposed for a class of MIMO discrete-time systems with multi-source disturbance, which

improves the defect of only traditional control method in terms o f the disturbance attenuation and

rejection. In the following, for a vector v(k), its Euclidean norm is denoted as v(k)=



(k)v(k)

and its L

norm is denoted as v(k)





∞

k=1

v(k)

2. FORMULATION OF THE PROBLEM

The following MIMO discrete-time system with nonlinear dynamics and two types of disturbances

is considered

x(k +1)= G

x(k)+ F

(x(k),k)+ H

[u(k)+d

(k)]+ H

(k)(1)

DOI: 10.1002/rnc

454

Int. J. Robust. Nonlinear Control

2012; :

–

22 453 472

COMPOSITE STRATIFIED ANTI-DISTURBANCE CONTROL

where x (k)∈ R

, u(k) ∈ R

(m<n) are the state, the control input, respectively. G

∈ R

n×n

, H

∈

n×

are the coefﬁcient matrices. H

∈ R

n×m

satisﬁes rank(H

)= m. F

is the corresponding

weighting matrices. f

(x(k),k) is nonlinear functions which are supposed to satisfy bounded

conditions described as Assumption 1. d

(k)∈ R

is supposed to be described by an exogenous

system with uncertainty in Assumption 2, which can represent the disturbance with partial known

information (e.g. constant and the harmonic signals). d

(k)∈ R



is the external disturbance in the

-norm. Such a model can also represent a wider class of controlled plants compared with [15].

Similar to [15], the following assumption is required to describe the nonlinear dynamics.

Assumption 1

The nonlinear functions satisfy

 f

(k),k)− f

(k),k)U

(k)− x

(k)), f

(0, k) = 0

(2)

where U

is known matrix.

Assumption 2

The disturbance in the control input path can be formulated by the following exogenous system:

w(k +1) = W w(k)+ H

(k)

(k) = V w(k)

(3)

where W ∈ R

r×r

, H

∈ R

r×l

and V ∈ R

m×r

are proper known matrices. (k) is the additional

bounded disturbance which result from the perturbations and uncertainties in the exogenous system.

Remark 1

In many cases, system disturbances can be described as a dynamic system with unknown parameter

and initial condition [2, 14, 16, 17]. For example, some disturbances can be described as harmonic

signal (periodic signal or sinusoidal signal) with known frequency and unknown amplitude and

phase. However, it has never been involved that uncertainties exist in such an exogenous model for

the disturbance, which conﬁnes the application of the results in [14, 16] (also in all other literature

where the exogenous system is assumed to have a precise model). Besides the other bounded

noises, dynamic system (3) can be used to describe the unmodeled error and system perturbations.

The following assumption is a necessary condition for the DOBC formulation.

Assumption 3

, H

) is controllable, (W, H

V ) is observable.

3. COMPOSITE STRATIFIED ANTI-DISTURBANCE CONTROL F OR THE CASE WITH

KNOWN NONLINEARITY

In this section, we suppose that f

(x(k),k) is given and Assumptions 1–3 hold. When all states

of the system are available, it is unnecessary to estimate the states, then the estimation of the

disturbance need to be concerned.

3.1. Disturbance observer

In this section, the disturbance observer is formulated as

(k +1) = (W + LH

V ) ˆw(k)+ L(G

x(k)+ F

(x(k),k)+ H

u(k))

ˆw(k) = (k)− Lx(k),

(k)= V ˆw(k)

(4)

DOI: 10.1002/rnc

455

Int. J. Robust. Nonlinear Control

2012; :

–

22 453 472

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