克服基于多重努斯鲍姆函数的稳定性定理的局限性

IF 1.8 4区 计算机科学 Q3 AUTOMATION & CONTROL SYSTEMS Mathematics of Control Signals and Systems Pub Date : 2024-09-09 DOI:10.1007/s00498-024-00400-w
Caiyun Liu, Yungang Liu
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引用次数: 0

摘要

在系统不确定性中,未知控制方向是一个相当重要的不确定性,其补偿应包含所谓的努斯鲍姆函数。最近,人们提出了基于多重努斯鲍姆函数(MNFs)的策略,以实现对具有非相同未知控制方向的系统的连续自适应控制,如大规模系统和多代理系统,而单一努斯鲍姆函数无法解决这些问题。但现有的稳定性定理要求MNF 有明确的表达式;MNF 都是奇数或都是偶数。本文旨在克服这些定理的局限性。我们首先从两个方面划分了 MNF 的基本性质,取代了文献中的显式表达。具体来说,MNFs 具有双变量-乘积形式,其中一个变量表示 MNFs 的符号频繁变化和持续强度,另一个变量表示 MNFs 的快速增长。这样的划分可以抓住 MNF 的本质,并尽可能多地涵盖 MNF。根据划定的 MNF,我们提出了两个稳定性定理,克服了上述局限性。值得注意的是,这两个定理并不要求 MNF 有明确的表达式。具体来说,定理 1 允许 MNFs 的非单调动态变量,同时要求 MNFs 为奇数,避免了过大的增益和过高的控制成本。定理 4 不要求所有 MNF 都是奇数,但要求 MNF 的动态变量必须是单调的,从而让用户在选择 MNF 时有更大的自由度。这两个稳定性定理分别应用于一个大规模系统和一个具有参数化不确定性的非线性系统,这两个系统都具有非相同的未知控制方向,在控制设计中分别避免了动态增益的单调性和过度参数化。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Overcoming limitations in stability theorems based on multiple Nussbaum functions

Among system uncertainties, unknown control direction is a rather essential one, whose compensation should entail the so-called Nussbaum function. Recently, strategies based on multiple Nussbaum functions (MNFs) have been proposed to make possible continuous adaptive control for the systems with nonidentical unknown control directions, such as large-scale systems and multi-agent systems, which cannot be addressed by a single Nussbaum function. But the existing stability theorems required: MNFs have explicit expressions; MNFs are all odd or all even. This paper aims to overcome the limitations of the theorems. We first delineate MNFs by two aspects of basic properties, replacing their explicit expressions as in the literature. Specifically, MNFs have the bivariate-product form, where one variate delineates frequent changes of signs and persistent intensity of MNFs, and the other delineates the rapid growth of MNFs. Such a delineation can capture the essence of MNFs and cover as many MNFs as possible. Based on the delineated MNFs, we present two stability theorems overcoming the aforementioned limitations. Notably, the two theorems do not require MNFs to have explicit expressions. Specifically, Theorem 1 allows nonmonotonic dynamic variables of MNFs while requiring MNFs to be odd, avoiding too large gains and excessive control cost. Theorem 4 does not require all MNFs to be odd while requiring dynamic variables of MNFs to be monotonic, giving users more freedom in selecting MNFs. The two stability theorems are applied to a large-scale system and a nonlinear system with parameterized uncertainties, both with nonidentical unknown control directions, to avoid monotonicity of dynamic gains and overparameterization, respectively, in control design.

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来源期刊
Mathematics of Control Signals and Systems
Mathematics of Control Signals and Systems 工程技术-工程:电子与电气
CiteScore
2.90
自引率
0.00%
发文量
18
审稿时长
>12 weeks
期刊介绍: Mathematics of Control, Signals, and Systems (MCSS) is an international journal devoted to mathematical control and system theory, including system theoretic aspects of signal processing. Its unique feature is its focus on mathematical system theory; it concentrates on the mathematical theory of systems with inputs and/or outputs and dynamics that are typically described by deterministic or stochastic ordinary or partial differential equations, differential algebraic equations or difference equations. Potential topics include, but are not limited to controllability, observability, and realization theory, stability theory of nonlinear systems, system identification, mathematical aspects of switched, hybrid, networked, and stochastic systems, and system theoretic aspects of optimal control and other controller design techniques. Application oriented papers are welcome if they contain a significant theoretical contribution.
期刊最新文献
Overcoming limitations in stability theorems based on multiple Nussbaum functions Stability analysis of systems with delay-dependent coefficients and commensurate delays Controllability with one scalar control of a system of interaction between the Navier–Stokes system and a damped beam equation On the relations between stability optimization of linear time-delay systems and multiple rightmost characteristic roots The local representation of incrementally scattering passive nonlinear systems
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