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

Lead Research Organisation: University of Kent
Department Name: Sch of Engineering & Digital Arts


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.

Planned Impact

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.
Description This research has developed new algorithms that have been used for the control of impact mechanical systems, particularly biped robots. In the case of biped robots, it is essential that algorithms converge in the finite time prescribed by the periods of time which determine the elements of the gait cycle. Without this, biped robots either are configured to walk very slowly or they fall over, with the number of falls increasing as the desired walking speed increases. Results for these finite time algorithms tested in France have been very good.
The research is also demonstrating that biological control systems within the human body which are essentially switched, can be analysed using the tools used by the switched control community. This is a reciprocal area, where engineering has much to learn from the robustness of the control schemes appearing in biology and where engineering analysis can assist in interpreting failure cases and transitions between health and disease. This research is also demonstrating that assessment of drug treatment is possible from a control perspective. The methods have been used to consider a range of conditions including HIV, Psoriasis and Measles.
Exploitation Route There would appear to be great potential to use these algorithms within autonomous manufacturing and other areas of process control where the ability to prescribe convergence robustly in a user specified finite time rather than asymptotically has the potential to improve efficiency and productivity.
To exploit the early findings in terms of the use of control theory for assessment of transitions between health and disease and the potential to analyse the effects of certain drug treatments requires cross disciplinary working across teams involving clinicians, control engineers, mathematical modellers, biologists.
Sectors Aerospace, Defence and Marine,Agriculture, Food and Drink,Healthcare,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology

Description A series of public lectures have been presented within the UK. These do not only demonstrate the classical engineering applications of control but also discuss biological control. This lecture seeks to inspire young people, particularly females who are under represented, to consider engineering as a fulfilling career
Sector Education
Impact Types Societal

Description The University of Sydney, Camperdown, New South Wales 2006, Australia 
Organisation University of Sydney
Country Australia 
Sector Academic/University 
PI Contribution Working in partnership with the University of Sydney, the tools developed in this project were used to investigate the feedback mechanism underlying the measles paradox. We determined mathematical conditions for the control and robustness of viral clearance. In particular, we helped characterize the importance of immunosuppression for viral clearance as well as the switch between T-cell mediated control and lymphocyte depletion. Our results are consistent with experimental data and help to untangle complex feedback mechanisms underlying similarities and differences in the dynamics of measles virus and canine distemper virus in their natural hosts. Together, our results advance understanding of virus-host interactions leading to the control of acute viremia and immunosuppression during morbillivirus infections.
Collaborator Contribution The partners provided biological expertise and fostered links to clinical collaborators.
Impact This partnership has developed a wider network of multidisciplinary collaborators including control engineers, mathematicians, biologists, clinical collaborators. Papers and a book Chapter have been produced.
Start Year 2017
Description Public lectures 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Public/other audiences
Results and Impact I have presented, and continue to present, public lectures which demonstrate how the methods we use to analyse switched control systems in engineering have a direct parallel with observations in biology. It is also shown how results from engineering can be used to understand health and disease.
Year(s) Of Engagement Activity 2019,2020,2021,2022