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

线性和非线性动力系统的有限时间镇定是当前国际上一个活跃的研究领域。在许多现有的工作有限时间收敛的李雅普诺夫函数的状态空间的原点是通过使用一个微分不等式，这是依赖于衰减率和已知的和不确定的系统参数的李雅普诺夫函数的增加条件。证明有限时间稳定性的基础上，这样一个强大的李雅普诺夫函数满足微分不等式提出了一个挑战相比，证明李雅普诺夫定理有关的渐近稳定性的考虑。当范例不仅需要稳定时间估计，而且还寻求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，任务可以进一步复杂化。最近的工作，在机械系统领域的调查人员已经获得了相应的结果，使用同质性的方法是建立在准同质性原则的可能不连续的系统，从而更广泛的范围内的不确定性比现有的文献。在初始数据和不确定性中统一的有限时间稳定性是可能的，这是使用现有方法无法保证的特征。建立时间的有限上界确定，而不需要找到一个满足微分不等式的李雅普诺夫函数。工作已经开发了一个单一的李雅普诺夫函数的不确定性，不连续的机械系统，以提供全球有限时间稳定性的系统的起源，在存在速度跳跃，而不必分析李雅普诺夫函数在跳跃瞬间，并已开发的参数化的滑模控制器，确保有限时间稳定的设计者指定的收敛时间和控制器参数被明确计算为一个所需收敛时间的函数。目前的概念证明表明，有限时间稳定性特征可以施加在可能的不连续系统，并提供了一个令人兴奋的平台，以探索当前感兴趣的更复杂的实际情况。显然，目前基于无限时间范围的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体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，经常发现不连续性。该项目旨在扩大系统类，其中发达的理论框架可以适用于包括这样的生物动力学。一个具体的驱动因素是参数化和评估免疫系统中存在的分叉，其中一个关键的范例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例来假设治疗以将系统移动回健康状态。自身免疫性疾病影响美国5000万人，是65岁以下女性的十大死亡原因之一，是慢性病的第二大原因，也是女性发病的首要原因。全世界自身免疫性疾病的病例数正在上升。这种受影响人数的增加和缺乏强有力的治疗方案导致自身免疫性疾病的发病率显着增加医疗保健支出以及劳动力生产力的损失，当然还有受影响者的生活质量差。目前还没有机制为基础的，概念性的理解自身免疫性疾病。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，这个问题。

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.