Stability and Robustness of Multivariable Feedback Systems

Abstract

This book on stability theory and robustness will interest researchers and advanced graduate students in the area of feedback control engineering, circuits, and systems. It will also appeal to mathematicians who are involved in applications of functional analysis to engineering problems. The book provides a methodology for the rigorous treatment of such inherently feedback aspects of dynamical system design as robustness and sensitivity, just as many researchers are beginning to realize that this type of methodology is mandatory if modern systems theory is to be used to design complicated multivariable and large-scale systems. The main objective of the book is to provide a clear mathematical formulation of the issues that arise in designing feedback systems that are robust against the destabilizing effects of unknown-but-bounded uncertainty in component dynamics. It is the first study to identify formal methods for the quantitative analysis of multiloop feedback system robustness. The view that is presents of nonlinear, multiloop feedback system stability theory is unique, lucid, and conceptually appealing. Lyapunov and input-output stability theories are unified in a new and simple geometrical perspective based on the topological separation of spaces. This perspective greatly facilitates visualization of the underlying conceptual issues in stability and robustness theory and serves to motivate specific results concerning the robustness of feedback systems. Potentially, this methodology may be applied to nonlinear feedback design, validation of modeling approximations, hierarchical control system design, and stability margin analysis for multiloop feedback systems. This book is the third publication in The MIT Press Series in Signal Processing, Optimization, and Control, edited by Alan S. Willsky.

Branch and bound algorithm with applications to robust stability

“…If α(A(δ)) = Re λ with λ = α + jω, then we search for ω in [0, 0.9 ω], and for ω in [2 ω, 20 ω]. In several cases (6,7,8,12,13,16,17,25,26,28,29) the value ω = 0 was obtained, in which event ω = 0 was chosen. In a number of cases the algorithm turns much faster with the frequency sweep method to reach the same precision.…”

“…A particularity of the three pruning tests (18), (20), (25) is that they may provide useful information even when the decision is P = not − pruning. Let ω 0 be a frequency such that μ (M( jω 0 )) < 1.…”

“…The principled mathematical tool proposed to describe these properties is the μ-singular value [31], which originated from work by Safonov [25] and Doyle [12]. Our present approach still owes much to this classical line as we use μ-upper bounds according to Fan et al [15] and Graham et al [16] to obtain conservative stability tests for evaluation, see Sect.…”

New results on H∞ filtering for fuzzy systems with interval time-varying delays

“…For instance, the kernel of the Lyapunov functional is chosen as piecewise linear in [12], which provides an alternative approach to the Lyapunov framework; the delay term is partitioned into two components in [17] to reduce conservatism and this methodology is further developed in [5]. In [11] the delay term is artificially partitioned to obtain less conservative stability criteria for time-delay systems, which is motivated by the topology separation approach proposed in [28] and its extension to the stability analysis of general linear systems in [15]. Taking advantage of the previous results [11], some results on the stability of time-delay fuzzy systems and their stabilization are obtained [46].…”

Stabilization and robustness of non-linear unity-feedback system: factorization approach

“…Stable non-linear S,(P, C) 2.1. Notation (Willems 1971, Desoer and Vadyasagar 1975, Safonov 1980 Let r c R and let v be a normed vector space. Let A feedback system is said to be well-posed if and only if for all possible inputs, all of the signals in the feedback system are determined by causal maps.…”