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

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.

This project will provide academic impact by developing new systems tools that can be applied to analyse complex systems which may exhibit discontinuous behaviour including bifurcations. This methodology will produce cross-disciplinary outcomes relevant to a range of biological systems, where biological switches occupy crucial decision-making points in the process of life and failure in the resulting control actions is often catastrophic. There is a need to train and develop researchers who can work at this intersection between such biological phenomenon and the mathematically motivated control and systems theory. Frequently such researchers as the PI from the domain of system control and observation have no advanced training in biology, given the educational entry requirements required to study mathematics and engineering within the UK at University level; similarly many experimental biologists have no training post-16 in mathematics. This project will enable the investigators, as well as the researchers working within the project, to develop expertise and knowledge within this increasingly important multi-disciplinary area. This in turn will provide mechanisms to study healthy and diseased systems and from this increased knowledge and understanding, impact in terms of improved treatment regimes as well as contributions to the development and assessment of drug treatments will follow. It is clear that systems analysis will never replace the need for laboratory experimentation and drug trials, but there is increasing evidence that modelling and analysis can inform laboratory experimentation and drug trials in a manner that saves both time and money. Some statistics relating to the incidence of immune disease as well as both the associated costs of treating the disease and developing new treatments clearly demonstrate the potential long term impact of research which seeks to develop a mechanism-based, conceptual understanding of autoimmune diseases. In the USA, approximately 50 million people suffer from autoimmune disease which represents approximately 20 percent of the population; the numbers affected are thus very large, and growing, and the impact on the quality of life and loss of earnings of these affected individuals is very large. In 2006, sources estimate that the amount spent worldwide on treatments or therapies for autoimmune disease was $18.3 billion where treatments for rheumatoid arthritis, multiple sclerosis, inflammatory bowel disease, and psoriasis - four of the most common autoimmune diseases - account for 75 percent of the autoimmune drug trials worldwide. There is thus a large and important economy associated with understanding of immune disease and work which can provide fundamental understanding of the surrounding issues also has the potential to create wealth and economic prosperity.