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，而不是渐近镇定，是当前国际上研究的一个活跃领域。在许多已有的工作中，Lyapunov函数有限时间收敛到状态空间的原点是利用由依赖于衰减率以及已知和不确定系统参数的微分不等式给出的关于该Lyapunov函数的递增条件来实现的。与关于渐近稳定性的Lyapunov定理的证明相比，基于这样一个满足微分不等式的强Lyapunov函数的有限时间稳定性的证明是一个挑战。当该范例不仅需要估计稳定时间，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，任务可能更加复杂。最近在机械系统领域的研究人员使用齐次方法获得了相应的结果，其中该方法建立在可能不连续系统的准齐次原理之上，因此允许比现有文献中的更大范围的不确定性。在初始数据和不确定性中一致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。在不需要寻找满足微分不等式的Lyapunov函数的情况下，确定了建立时间的有限上界。工作已经为不确定的、不连续的机械系统开发了单一的Lyapunov函数，以便在存在速度跳跃的情况下为系统的原点提供全局有限时间稳定性，而不必分析跳跃瞬间的Lyapunov函数，并且已经开发了确保有限时间稳定的滑模控制器的参数化，其中设计者指定了收敛时间，并且控制器参数被显式地计算为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不连续的系统中，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，当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，个体克隆性免疫细胞群需要在有限的时间内扩张和激活。在自然界更远的地方，由于进化，经常会发现不连续的现象。该项目旨在扩大所开发的理论框架可以应用于的系统类别，以涵盖此类生物动力学。一个特定的驱动因素是对免疫系统中存在的分歧进行参数化和评估，其中一个关键范式是调查触发事件如何将免疫系统从健康状态转移到自身免疫状态，以及如何使用控制范式来假定治疗将系统恢复到健康状态。在美国，自身免疫性疾病影响着5000万人，是65岁以下女性死亡的十大原因之一，是慢性疾病的第二大原因，也是女性发病率最高的原因。全球自身免疫性疾病的病例数量正在上升。受影响人数的增加和缺乏强有力的治疗机制导致了自身免疫性疾病的发生，大大增加了医疗支出，降低了劳动力的生产力，当然也降低了受影响者的生活质量。目前对自身免疫性疾病还没有基于机制的概念性理解。这个项目寻求发展和应用新兴的方法，从有限时间镇定不确定的可能不连续的动态系统来解决这个问题。

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.