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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

输入非线性和控制方向未知MIMO系统的渐

近模糊控制

陈辞，刘治，章云

广东工业大学自动化学院，广州　510006

摘要：本文针对未知相同控制方向和输入非线性的MIMO非线性系统提出了一种渐近模糊控制

方法。通过构造一种新型的Nussbaum定理，解决了来自输入非线性及其符号的控制难题。得

益于该定理，未建模有界系统动态和逼近误差亦可在没有额外构造鲁棒项的基础上完成分析。

因此，所提的基于Nussbaum函数控制方法，不但具有设计上简单的特点，而且具有控制鲁棒

性。进一步地，所提模糊控制方案将传统的半局域一致最终有界结论从有界误差提升为渐近控

制。最后，仿真例子说明了所提方法的有效性。

关键词：自动控制技术；输入非线性；未知控制方向；渐近控制

中图分类号： TP13

Asymptotic Fuzzy-approximation based Control

of MIMO Systems with Unknown Input

Nonlinearities and Control Direction

CHEN Ci, LIU Zhi, ZHANG Yun

School of Automation, Guangdong University of Technology, Guangzhou 510006

Abstract: This paper investigates a fuzzy asymptotic control for MIMO systems with

unknown identical control directions, input nonlinearities and external disturbances. Control

diﬃculties from input nonlinearities and their control signs in MIMO systems are

simultaneously tackled with the help of a newly constructed Nussbaum theorem. Beneﬁting

from this theorem, unmodelled bounded system dynamics and universal approximation errors

are also handled without constructing extra robust terms. Hence, both the control simplicity

and control robustness are obtained within the frame of the developed Nussbaum tool.

Furthermore, under the semiglobally uniformly ultimately bounded condition, the proposed

fuzzy control scheme has succeeded in extending the bounded result to the asymptotic

convergence. Finally, a simulation example is carried out to show the proposed approach’s

eﬀectiveness.

基金项目： Specialized Research Fund for the Doctoral Program of Higher Education (20124420130001)

作者简介： Chen Ci（1989-），male，Ph.D candidate，major research direction：adaptive control, robotic control.

Correspondence author：Liu Zhi（1977-），male，professor，major research direction：adaptive control, robotics,

intelligent controls. Zhang Yun (1963-)，male，professor，major research direction：multi-agent system control,

robotics, intelligent controls.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Key words: Automatic control; input nonlinearities; unknown control direction; asymptotic

control

0 Introduction

It is reported that [1] is the ﬁrst to investigate a class of ﬁrst-order linear systems with

unknown control direction problem. A special function based control approach was developed

in [1], known as the Nussbaum gain. Soon afterwards, the Nussbaum gain was successfully

integrated with the adaptive control to solve the control direction problem for ﬁrst-order non-

linear systems [2] and for Single Input Single Output (SISO) systems with unknown constant

coeﬃcients[3, 4, 5]. Subsequently, Ge et al. [6] creatively dealt with unknown time-varying un-

certain coeﬃcients and unknown bounded uncertainties for a class of SISO nonlinear systems

by proposing a new Nussbaum gain based technical lemma. It is worthy noting that aforemen-

tioned results are carried out based on the assumption that the investigated systems belong to

a class of SISO systems. However, when the systems are extended to Multiple Input Multiple

Output (MIMO), the control design and stability analysis procedure are not an exact copy of

the case for SISO systems.

The major concern arises that the rationality is not ensured theoretically when the dimen-

sion of the Nussbaum gain is extended from one to multiples. In order to overcome this restric-

tion, some pioneering works [7, 8] have used the decentralized concept and have decomposed

MIMO systems into several separate SISO systems. Consequently, the classical Nussbaum gain

is employed to identify the control direction of each SISO subsystem [9]. Currently, Chen et

al. [10] successfully proved that multiple Nussbaum gains could be analyzed in one Lyapunov

function candidate and thus proposed a promising analysis tool based on a novel Nussbaum

gain. Moreover, [11] theoretically extended the Nussbaum approach from unknown constant

coeﬃcients [10] to unknown time-varying ones. Therefore, the communication bridge between

the actuator nonlinearities and control direction was built in [11]. Most recently, Ding [12] took

a further step forward to remove the prior knowledge of upper and lower bounds of the constant

coeﬃcients, and constructed the promising scheme to realize the output control of multi-agent

system. Subsequently, to eliminate the control shock from the traditional Nussbaum gains, a

new type of saturated Nussbaum gain for MIMO systems has currently developed with the

idea of Time-Elongation in [13]. Though progresses have been achieved, the current Nussbaum

approach is limited to the situation where disturbances are ignored and coeﬃcients are constan-

t. It is noted that unknown external uncertainties and time-varying coeﬃcients are inventible

factors in the modelling procedure [14, 15, 16, 17, 18, 19, 20, 21, 22]. Without considering these

unknown characteristics into control schmoe, the control failure or instability will be generat-

ed in the highly demanding areas. Hence, it is still an important task to show whether the

- 2 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Nussbaum gain based approach is robust to unknown external uncertainties and time-varying

coeﬃcients in MIMO systems.

Motivated by above analyses, a novel robust Nussbaum tool is developed to tackle the

control problem for MIMO systems with unknown identical control directions. The main con-

tributions of this paper are summarized as: 1). With the help of the newly-constructed Nuss-

baum tool, a promising analysis framework is established to pave the way for tackling unknown

uncertainties, identical control directions and timevarying control coeﬃcients, simultaneously.

2). Moreover, it is rigourously proved that the proposed tool can be combined with adaptive

fuzzy control such that the conventional result from semiglobally uniformly ultimately bound-

ed condition is extended. Therefore, within the frame of the proposed control, the asymptotic

convergence is guaranteed without constructing extra robust terms. The remaining parts of

this paper are organised as follows. In Section II, brief review and problem formulation are

stated. In Section III, the introduction to fuzzy approximation based control is presented. In

Section IV, an adaptive fuzzy control scheme is developed for MIMO systems; and its stability

analysis is presented. In Section V, The simulation is carried out to evaluate the eﬀectiveness

of the proposed control. In Section VI, the conclusion is contained.

1 Main Results and Proofs: Robust Nussbaum Gain

Based Approach

A key theorem which facilitates the stability analysis for the control problem form unknown

control directions, input nonlinearities and external disturbances is presented as follows.

Theorem 1. Given that g

(t) is an unknown time-varying function, which satisﬁes g

(t) ∈

∇ = [g

−

, g

] with 0 /∈ ∇, g

−

, min

1≤k≤n

(t)} and g

, max

1≤k≤n

(t)}. Let V

, t) and ζ

(t) be

smooth functions deﬁned on the time interval [t

, t

) with V (t

, t) being nonnegative and ζ

)

being bounded. Moreover, N

(ζ

(t)) is deﬁned as

(ζ) = e

(ζ

+ 2) sin(ζ). (1)

If the following inequality holds:

V (t

, t) ≤

k=1

−α

(t)N

(ζ

(%))

(%)e

k=1

−α

(%)e

d% + λ,

(2)

where α

is a positive constant and λ is a bounded variable, then the conclusion is drawn that

(t), V (t

, t), e

−α

(%)e

d% and e

−α

(t)N

(ζ

(%))

(%)e

d% must be bounded

on the interval [t

, t

) for k = 1, 2, ..., n.

- 3 -

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