Low gain magic: Control of linear systems with distributed infinite input delays

Time delays are intrinsic features for various practical systems in engineering, biology, chemistry, economics, mechanics, physics, physiology, population dynamics, and so on. Unfortunately, systems with time delays are difficult to deal with because they belong to the infinite dimensional differential functional equations. Most of existing works on time-delayed systems focus on bounded time-delays, and various approaches to stability analysis and controller synthesis have been developed. Recently, there are increasing interests in systems with infinite delays as infinite delays indeed exist in some real-world systems, such as the HIV spread model, the oscillator model, the traffic flow model, the neural network model, and so on. Infinite delays, also known as unbounded delays, are more general but also more difficult to deal with. Infinite delays can be classified as either time-varying infinite delays or distributed infinite delays. The major challenges in dealing with infinite delays, especially in control of systems with infinite delays, include limitations of analysis and synthesis tools, the sensitivity of solutions to initial conditions, and mathematical complexity.

A group of researchers at the City University of Hong Kong, Department of Biomedical Engineering: Dr. Xiang Xu, Dr. Lu Liu and Professor Gang Feng investigated the stabilization problem of systems with distributed infinite delays at the input. They developed two low gain feedback controllers, for two different classes of linear systems with distributed infinite input delays. They considered two cases where the first case comprised of unstable eigenvalues at the imaginary axis while the second case consisted of unstable eigenvalues at the origin. Furthermore, they showed that the proposed low gain controllers could be used to stabilize those linear systems with a distributed infinite input delay. Their research work is published in the research journal, Automatica.

The authors observed that the linear system could be globally stabilized asymptotically by using the low gain controllers. Consequently, the use of low gain feedback in solving stabilization problems of linear systems with distributed infinite delays at the input was efficiently confirmed. This was attributed to the fact that the results were based on the stability results of the systems with infinite delays. Furthermore, unlike in the previous studies, the authors noted that the existing results in the literature concerning the bounded distributed input delays as well as constant input delays were both considered special with regard to the new results obtained.

The study by City University of Hong Kong scientists is the first to successfully solve stabilization problems of linear systems with distributed infinite input delays based on low gain feedback controllers. The effectiveness and efficiency of the proposed low gain controllers are validated through numerical simulations in two engineering systems. Their new method provides a powerful and useful basis in further study of systems with infinite delays, and also provides a great potential for their applications in real-world systems.

P.S The authors acknowledge the support by the research Grants Council of Hong Kong under grant CityU-11206817.