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Applied Math Seminar
Delay effects on stability. A robust control approach
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The delay systems (called also hereditary or systems with aftereffects) represent a class of infinite-dimensional systems largely used to describe propagation phenomena or population dynamics. Roughly speaking, the reaction of real world systems to exogenous signals is never instantaneously and it needs some time, time which can be translated into a mathematical language by some delay terms. A distinguished feature of this class of systems is that their evolution rate is described by differential equations which include information on the past history. Into a mathematical framework, such systems may be described in several ways, and we mention, for example, differential equations on abstract spaces, over rings of operators or functional differential equations. In system theory, we may use infinite-dimensional, 2D or behavioral based representations. Independently of the representation type, the delay effects on the stability and con- trol of dynamical systems (delays in the state and/or in the input) are problems of recurring interest since the delay presence may induce complex behaviors (oscillations, in- stability, bad performances) for the (closed-loop) schemes: small delays may destabilize some systems, but large delays may stabilize others. In control, a lot of results pointed out that delays in feedback systems are accompanied by bandwidth sensitivity to model uncertainty. Furthermore, delay perturbations due to some modeling errors may induce instability, and interconnection schemes of Þnite or infinite-dimensional systems with delay blocks may become unstable even if some well-possedeness property is verified. All these aspects motivate the study of delay effects on (closed- loop) dynamical systems properties. The aim of this talk is to present briefly some user-friendly methods and tech- niques for the analysis and control of the dynamical systems in presence of delays in frequency-domain, approaches based on a particular interpretation of delays in a robust control framework. The presentation is as simple as possible, focusing more on the main intuitive ideas to develop theoretical results, and their potential use for practical applications. Single and multiple delays will be both considered. A special attention will be paid to the geometry of stability regions in the delay parameters space. Finally, two applications will be presented. The first example is represented by a 1 synchronization scheme to achieve a high level of consistency in peer-to-peer based virtual environments for shared haptics with large delay. The synchronization scheme makes use of an advanced feedback controller to compensate for the state error between geographically separated sites. The maximum allowable (communication) delay will be explicitly computed. The second example is a dynamical delay system encountered in modeling the post-transplantation dynamics of the immune response to chronic myelogenous leukemia. Such models include multiple delays in large range, from one minute to several days. The main objective is to understand the interactions between small and large delays in deÞning the stability regions in the delay parameter space. |