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Finite time orbitally stabilizing synthesis of complex dynamic systems with bifurcations with application to biological systems

具有分岔的复杂动态系统的有限时间轨道稳定合成及其在生物系统中的应用

基本信息

批准号：
EP/J018295/1
负责人：
Sarah Spurgeon
金额：
$ 41.98万
依托单位：
University of Kent
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ018295%2F1
关键词：
Finite time orbitally stabilizing synthesis

项目摘要

Stabilization, in finite time rather than asymptotically, of linear and non-linear dynamical systems is an active current area of research internationally. In much of the existing work finite time convergence of a Lyapunov function to the origin of the state space is achieved using an increasing condition on that Lyapunov function given by a differential inequality which is dependent upon the decay rate and both known and uncertain system parameters. The proof of finite time stability on the basis of such a strong Lyapunov function satisfying a differential inequality poses a challenge when compared to proofs of Lyapunov theorems relating to asymptotic stability considerations. The task can be further complicated when the paradigm requires not only a settling time estimate but also seeks to achieve parameter selections for a control strategy to ensure an apriori chosen settling time is achieved. Recent work by the investigators in the domain of mechanical systems has obtained corresponding results using a homogeneity approach where the methodology is founded on a quasihomogeneity principle of possibly discontinuous systems, and thus a broader range of uncertainty is permitted than in the existing literature. Finite time stability which is uniform in the initial data and in the uncertainty is possible, a feature that cannot be guaranteed using existing methods. A finite upper bound on the settling time is determined without the need to find a Lyapunov function satisfying a differential inequality. Work has developed a single Lyapunov function for uncertain, discontinuous mechanical systems to provide global finite time stability to the origin of the system in the presence of velocity jumps without having to analyze the Lyapunov function at the jump instants and has developed parameterisations of sliding mode controllers that ensure finite time stabilisation where the designer specifies a convergence time and controller parameters are explicitly computed as a function of the required convergence time. The current proof of concept demonstrates that finite time stability characteristics can be imposed in possibly discontinuous systems and provides an exciting platform to explore more complex practical scenarios of current interest. It is clear that current methods which analyse systems based upon an assumption of an infinite time horizon are frequently flawed. For example, individual clonal immune cell populations are required to expand and become activated for limited time. Further in the natural world, discontinuity is frequently found as a result of evolution. This project seeks to broaden the system class to which the developed theoretical framework can be applied to encompass such biological dynamics. One specific driver is to parameterise and assess the bifurcations present in the immune system, where a key paradigm is to investigate how a triggering event may move the immune system from the healthy to the autoimmune state and also how control paradigms can be used to postulate treatment to move the system back to the healthy state. Autoimmune disease affects 50 million people in the USA where it is one of the top ten causes of death in women under 65, is the second highest cause of chronic illness, and is the top cause of morbidity in women. The number of cases of autoimmune disease are rising across the world. This rise in the number of people affected and the absence of robust treatment regimes results in the incidence of autoimmune disease contributing significantly to the rise in healthcare spending as well as loss of productivity in the workforce and of course poor quality of life for those affected. There is currently no mechanism-based, conceptual understanding of autoimmune disease. This project seeks to develop and apply emerging methods from finite time stabilisation of uncertain possibly discontinuous dynamic systems to this problem.

线性和非线性动力系统的有限时间而非渐近稳定是目前国际上研究的一个活跃领域。在许多现有的工作中，李雅普诺夫函数的有限时间收敛到状态空间的原点是使用由微分不等式给出的李雅普诺夫函数的递增条件来实现的，该条件依赖于衰减率和已知和不确定的系统参数。与关于渐近稳定性的李雅普诺夫定理的证明相比，基于满足微分不等式的这种强李雅普诺夫函数的有限时间稳定性的证明是一个挑战。当范式不仅需要估计稳定时间，而且还需要实现控制策略的参数选择以确保实现先验选择的稳定时间时，任务可能会进一步复杂化。最近在机械系统领域的研究人员使用同质性方法获得了相应的结果，该方法建立在可能不连续系统的准同质性原理上，因此允许比现有文献更广泛的不确定性范围。有限时间稳定性在初始数据和不确定性中是一致的，这是现有方法无法保证的特征。在不需要寻找满足微分不等式的李雅普诺夫函数的情况下，确定了稳定时间的有限上界。工作人员开发了一个李雅普诺夫函数，用于不确定，不连续机械系统，在存在速度跳跃的情况下为系统原点提供全局有限时间稳定性，而无需分析跳跃瞬间的李雅普诺夫函数，并开发了滑模控制器的参数化，确保有限时间稳定性，其中设计者指定收敛时间，控制器参数作为所需收敛时间的函数显式计算。目前的概念证明表明，有限时间稳定性特征可以在可能的不连续系统中施加，并为探索当前感兴趣的更复杂的实际场景提供了一个令人兴奋的平台。很明显，目前基于无限时间范围假设来分析系统的方法经常是有缺陷的。例如，单个克隆免疫细胞群需要扩增并在有限的时间内被激活。此外，在自然界中，由于进化的结果，经常发现不连续性。该项目旨在扩大系统类，使已开发的理论框架可以应用于涵盖这种生物动力学。一个特定的驱动因素是参数化和评估免疫系统中存在的分支，其中一个关键范式是研究触发事件如何将免疫系统从健康状态移动到自身免疫状态，以及如何使用控制范式来假设治疗以将系统移回健康状态。自身免疫性疾病在美国影响着5000万人，是65岁以下妇女死亡的十大原因之一，是慢性疾病的第二大原因，也是妇女发病的首要原因。在世界范围内，自身免疫性疾病的病例数量正在上升。受影响人数的增加以及缺乏强有力的治疗制度导致自身免疫性疾病的发病率显著增加，导致医疗保健支出增加，劳动力生产力下降，当然还有受影响者的生活质量下降。目前对自身免疫性疾病还没有基于机制的概念性理解。该项目旨在开发和应用新兴的方法，从有限时间稳定的不确定可能不连续的动态系统来解决这个问题。