On the relations between stability optimization of linear time-delay systems and multiple rightmost characteristic roots

IF 1.8 4区 计算机科学 Q3 AUTOMATION & CONTROL SYSTEMS
Wim Michiels, Silviu-Iulian Niculescu, Islam Boussaada, Guilherme Mazanti
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引用次数: 0

Abstract

Several recent results on spectrum-based analysis and control of linear time-invariant time-delay system concern the characterization and exploitation of situations where the so-called multiplicity-induced dominancy property holds, that is, the higher multiplicity of a characteristic roots implies that it is a rightmost root. This direction of research is inspired by observed multiple roots after minimizing the spectral abscissa as a function of controller parameters. However, unlike the relation between multiple roots and rightmost roots, barely theoretical results about the relation of the former with minimizers of the spectral abscissa are available. Consequently, in the first part of the paper the characterization of rightmost roots in such minimizers is briefly revisited for all second-order systems with input delay, controlled with state feedback. As the main theoretical results, the governing multiple root configurations are proved to correspond not only to rightmost roots, but also to global minimizers of the spectrum abscissa function. The proofs rely on perturbation theory of nonlinear eigenvalue problems and exploit the quasi-convexity of the spectral abscissa function. In the second part, a computational characterization of minima of the spectral abscissa is made for output feedback, yielding a more complex picture, which includes configurations with both multiple and simple rightmost roots. In the analysis, the pivotal role of the invariant zeros is highlighted, which translate into restrictions on the tunable parameters in the closed-loop quasi-polynomial.

Abstract Image

论线性时延系统的稳定性优化与多最右特征根之间的关系
最近关于基于频谱的线性时变时延系统分析和控制的几项成果涉及到所谓的多重性诱导支配特性(即特征根的多重性越高,意味着它是最右边的根)成立情况的特征描述和利用。这一研究方向的灵感来自于在最小化作为控制器参数函数的频谱缺省值后观察到的多重根。然而,与多根和最右根之间的关系不同,前者与频谱尾数最小化之间的关系几乎没有理论结果。因此,本文第一部分将简要重述所有带输入延迟、用状态反馈控制的二阶系统中最右根的特征。作为主要的理论结果,本文证明了治理多根配置不仅对应于最右根,而且对应于谱离差函数的全局最小值。证明依赖于非线性特征值问题的扰动理论,并利用了频谱Abscissa函数的准凸性。在第二部分中,对输出反馈的频谱尾数最小值进行了计算表征,得出了一个更为复杂的图景,其中包括具有多个最右根和简单最右根的配置。在分析中,我们强调了不变零点的关键作用,它转化为对闭环准多项式中可调参数的限制。
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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.
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