基于分布式反演的高阶非线性动力学自适应模糊控制方法研究

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"基于分布反演的多个高阶非线性动力学自适应模糊控制" 本文的主要目的是研究多个高阶非线性动力学的协同自适应模糊控制。为了实现这一目标，我们首先对通信网络拓扑进行了分析，发现其为无向的和固定的。然后，我们对每个单独的动力学进行建模，使用高阶积分器来描述其未知的非线性动力学和未知的外部干扰。在此基础上，我们引入了模糊逻辑系统，以补偿未知的非线性动力学。通过自适应模糊逻辑系统方案，我们可以对未知的非线性动力学进行补偿，并抵消近似误差和外部干扰的负面影响。为了实现分布式控制，我们设计了两个协作自适应模糊控制器，分别对应于每个代理。在backstepping框架下，我们设计了分布式控制器，以使所有代理最终达成共识。这些控制器只需要代理及其邻居之间的本地状态信息，因此它们是分布式的。最后，我们通过一个具有四阶动力学的仿真示例证明了算法的有效性。本文的主要贡献在于提出了一种基于分布反演的多个高阶非线性动力学自适应模糊控制方法，该方法可以对多个高阶非线性动力学进行协同控制，并且具有鲁棒性和自适应能力。知识点： 1. 高阶非线性动力学建模：使用高阶积分器对未知的非线性动力学和未知的外部干扰进行建模。 2. 模糊逻辑系统：引入模糊逻辑系统以补偿未知的非线性动力学。 3. 自适应模糊逻辑系统方案：使用自适应模糊逻辑系统方案对未知的非线性动力学进行补偿。 4. 分布式控制：设计分布式控制器，以使所有代理最终达成共识。 5. Backstepping框架：使用backstepping框架来设计分布式控制器。 6. 协作自适应模糊控制：设计两个协作自适应模糊控制器，分别对应于每个代理。 7. 仿真示例：通过一个具有四阶动力学的仿真示例证明了算法的有效性。技术关键词： * 多个高阶非线性动力学 * 协同自适应模糊控制 * 分布式控制 * Backstepping框架 * 模糊逻辑系统 * 高阶积分器 * 未知的非线性动力学 * 未知的外部干扰 * 近似误差 * 鲁棒性项这篇论文的主要贡献在于提出了一种基于分布反演的多个高阶非线性动力学自适应模糊控制方法，该方法可以对多个高阶非线性动力学进行协同控制，并且具有鲁棒性和自适应能力。这篇论文的结果可以应用于多个领域，如机器人学、自动化控制、智能系统等。

Distributed backstepping-based adaptive fuzzy control of multiple high-order nonlinear dynamics 65

the Brunovsky form with uncertainties is considered,

which include ﬁrst- and second-order systems as spe-

cial cases. Second, a systematic controller design pro-

cedure is proposed to deal with the control problem

through combining distributed backstepping method

with adaptive fuzzy control techniques. The adaptive

fuzzy control is completely distributed. And the conver-

gence of the system errors is proved rigorously by virtue

of the Lyapunov stable theory and Barbalat’s lemma.

The subsequent sections are organized as follows: In

Sect. 2, the control problem is formally stated and the

background as well as necessary preliminaries concern-

ing the control problem are given. In Sect. 3, the cooper-

ative control laws are proposed relying on backstepping

method and adaptive fuzzy control approaches. The

unknown nonlinear functions are dealt by fuzzy logic

systems, and the external disturbances are addressed

by applying robust adaptive control method. In Sect. 4,

a four-order simulation example is provided to demon-

strate the performance of the proposed control laws.

The last section concludes this paper.

2 Preliminaries and problem statement

In this section, basic graph theory for networked

dynamics, control problem and fuzzy logic systems on

graph are introduced.

2.1 Brief graph theory for networked dynamics

With respect to networked dynamics, any control laws

must be distributed in the sense that it respects the

communication network topology. The communication

restrictions by topologies can severely limit the power

of local distributed control algorithm at each individ-

ual dynamics. The idea of a communication network

models the information ﬂows in a multi-agent group.

A team of m high-order nonlinear dynamics labeled

as system 1 to m are considered. The communica-

tion topology among the m systems is assumed to be

bidirectional or directional, and the interactions among

the nodes are represented by an undirected or directed

graph G = (V, E, A). The topology G represents the

structure of networked system, where V is a set of the

indices of the systems and E ⊆ V × V is a set of edges

that describe the communications between the s ystems.

If ( p , j ) ∈ E, then p is neighboring to j, meaning sys-

tem j can obtain information from system p. A is a

weighted adjacency matrix with nonnegative adjacency

elements a

. Moreover, it is assumed that a

= 0. If

the state of system p is available to system j, then sys-

tem p is said to be a neighbor of system j. The neighbor

set of node v

is denoted by N

, where j /∈ N

[25].

2.2 Problem statement

In practice, there are several cases where the individual

dynamics can be described by a higher-order form. For

example, the jerk system is described by third-order

differential equations [1]. A single-link ﬂexible joint

manipulator can be modeled by a fourth-order nonlin-

ear system. A group of unmanned aerial vehicle forma-

tion control problem, in essence, is a high-order multi-

agent system network coordination problem. In partic-

ular, the features of high order is more obvious during

the aircraft through tactical maneuvers. In addition, due

to the imprecision measurement and interactions with

complex environments, networked nonlinear systems

with uncertainties and external disturbances have to

be investigated simultaneously in practice. For those

complex nonlinear dynamics, the Brunovsky canoni-

cal form can been obtained through the linearization

method. In the process of model transformation, the

unmodeled dynamics and disturbances are embodied

in the smooth nonlinear function and external distur-

bances.

Take the single-link ﬂexible joint manipulator as a

representative example, whose dynamics can be written

as [15]

˙x

= x

, ˙x

=−

MgL

sinx

−

− x

˙x

= x

, ˙x

− x

) +

where I , J are, respectively, the link and the rotor iner-

tia moments, M is the link mass, k is the joint elastic

constant, L is the distance from the axis of the rota-

tion to the link center of mass and g is the gravitational

acceleration, respectively. This nonlinear dynamics can

be transformed to the higher-order normal form with

= x

˙z

= z

, ˙z

= z

, ˙z

= z

, ˙z

= a(z) + b(z)u,

In this paper, we consider a group of m (m ≥ 2)

systems with non-identical dynamics distributed on

an undirected communication network G. The dynam-

ics of the j-th system is described in the nonlinear

Brunovsky form as

123

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