自适应模糊控制：切换非线性系统的观测器设计与稳定性分析

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"任意切换下非线性系统基于观测器的自适应模糊控制" 这篇研究论文主要探讨了在任意切换情况下的非线性系统的自适应模糊控制问题。在现代工程领域，如机器人、机械设备等，切换系统有着广泛的应用。因此，对这类系统的控制策略研究受到了广泛关注。传统的控制方案通常侧重于状态反馈，而对于切换线性或非线性系统，已有不少文献进行了报告。论文提出了一种基于观测器的自适应模糊控制方法，专门针对一类非线性切换系统。其中，设计了一个切换模糊观测器来估计无法直接测量的系统状态。这一观测器能够有效地跟踪系统状态，即使在系统切换过程中也能保持稳定性和准确性。关键创新点在于使用切换Lyapunov函数进行控制设计和稳定性分析。Lyapunov函数是系统稳定性分析中的核心工具，通过构造合适的切换Lyapunov函数，可以确保控制系统在不同工作模式间切换时保持稳定性。此外，论文还提出了一种基于反向传播的自适应模糊输出反馈控制策略。该策略利用反向传播技术逐步设计控制器，以确保所有闭环信号保持有界，并且跟踪误差能收敛到原点附近的一个小邻域。这表明所提出的控制策略能够在任意切换条件下实现良好的控制性能和跟踪精度。在非线性切换系统中，模糊逻辑被用来近似复杂的非线性关系，这是因为模糊逻辑系统具有强大的非线性建模能力，并且易于实现自适应调整。通过自适应算法，模糊系统的规则参数可以在线调整，以适应系统动态特性的变化。该论文为解决非线性切换系统的控制问题提供了一个新颖而有效的解决方案，不仅考虑了系统状态的不可测性，还解决了系统在不同工作模式间的稳定性问题。这一研究对于实际工程应用，尤其是那些涉及复杂非线性和动态切换的系统，具有重要的理论和实践价值。

Observer-based Adaptive Fuzzy Control of Switched Nonlinear Systems

Under Arbitrary switchings*

Zhiliang Liu

, Bing Chen

, Chong Lin

, and Li Zhang

Abstract— This paper addresses the problem of output feed-

back adaptive fuzzy control for a class of nonlinear switched

systems. A switched fuzzy observer is constructed to estimate

the unmeasurable system state. Meanwhile, switched Lyapunov

candidate functions are applied to control design and stability

analysis. Furthermore, a backstepping-based adaptive fuzzy

output feedback control strategy is proposed. It is shown that

the suggested controllers guarantee that all the closed-loop

signals remain bounded and the tracking error converges to

a small neighbourhood around the original point.

I. INTRODUCTION

Switched systems can be found important applications in

practice engineering, such as robotic, mechanical systems,

and so on [1] and [2]. So, the control issues of switched

systems have been paid great attentions in the last decade.

Some control schemes via state feedback have been reported

for switched linear or nonlinear systems, respectively, for

instance see [3]-[8]. In practice engineering, system’s state

variables are usually unmeasurable. In this case, those control

strategies via state feedback become infeasible. Recently,

adaptive fuzzy/neural control technique has been applied to

solve output feedback control problems of switched uncertain

nonlinear systems. The work in[9] addressed the problem of

output feedback stabilization for switched nonlinear uncer-

tain systems in strict feedback form. An linear observer was

set up to estimate the unknown system’s state and radial basis

function (RBF) neural networks (NNs) were employed to ap-

proximate those unknown nonlinear functions. Furthermore,

an observer-based adaptive neural backstepping controller

design procedure was proposed in [10]-[13]. The devel-

oped adaptive neural controller ensured that all the closed-

loop signals were uniformly ultimately bounded. Differing

from the work in [9], a fuzzy observer was constructed

for switched nonlinear systems to estimate the unknown

system state in [12]. Fuzzy logic systems were introduced

in the design observer to compensate for the nonlinearities

in the controlled systems. Furthermore, an adaptive output

feedback fuzzy control scheme was proposed. The suggested

controllers guarantee the closed-loop stability. Since fuzzy

logic systems are introduced in observer to compensate for

the system nonlinearity, the proposed state estimation in

[12] can get smaller estimation errors than that in [9]. This

result was further extended to switched nonlinear larger-scale

*This work was supported by the National Natural Science Foundation

of China under Grant Nos. 61473160, 61673227

Z.L. Liu, B. Chen, C. Lin ad L. Zhang are all with the Insti-

tute of Complexity Science, Qingdao University, Qingdao 266071, China

hartright@foxmail.com

systems in [13], in which an adaptive fuzzy decentralized

tracking control strategy was presented via output feedback.

Even though some results on observer-based adaptive

fuzzy/neural control for switched nonlinear systems have

been reported, those existing results are not perfect. Some

weakness of the aforementioned results need to be fur-

ther modiﬁed. For examples, 1) in the existing adaptive

fuzzy/neural control schemes based on linear observer, the

system nonlinearities are not well compensated for. So,

these nonlinearities may lead to the deterioration of the

estimation precision; 2) in the control schemes proposed

via fuzzy/neural observers, the observer gain matrix L is

obtained by solving the Lyapunov matrix equation (A −

LC)

P + P (A − LC) + Q = 0. Meanwhile, the stability of

the observation error dynamics requires the positive deﬁnite

matrices P and Q satisfying a nonlinear matrix inequality

min

(Q) − a||P ||

− n − 2 > 0, where λ

min

(Q) is the

minimum eigenvalue of Q, a is a positive constant regarding

a sum of some Lipschitz constants and n denotes the order

of the systems. Usually, it is difﬁcult to ﬁnd positive deﬁnite

matrices P and Q and matrix L simultaneously satisfy

the Lyapunov matrix equation and the nonlinear matrix

inequality.

Based on the above observation, this paper still studies

adaptive fuzzy output feedback tracking control for a class

of switched nonlinear systems in strict-feedback form. A

fuzzy observer is used to estimate the unknown system state.

Then, differing from the approach presented in [12] and [13],

convex combination technique is used to get the observer

gain matrix. So, our approach avoids to solve a nonlinear

matrix inequality. The above mentioned weaknesses of the

existing results have been overcome. It is shown that the sug-

gested controllers guarantee that all the closed-loop signals

remain bounded and the tracking error converges to a small

neighbourhood around the original point.

II. PRELIMINARIES AND PROBLEM STATEMENT

Consider the following switched nonlinear system in strict

feedback form:

˙x

= f

i,σ( t)

(¯x

) + x

i+1

˙x

= f

n,σ (t)

(¯x

) + u

σ (t)

y = x

, (1)

where x

∈ R is the state of the ith subsystem and ¯x

, x

. . . , x

]

, σ(t) : R

→ M = {1, 2, . . . , m} denotes

a switching signal. Particularly, it can be expressed as σ(t) =

k(k ∈ M), a piecewise continuous-time function. f

i,σ( t)

(1 ≤

2017 11th Asian Control Conference (ASCC)

Gold Coast Convention Centre, Australia

December 17-20, 2017

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