Research overview
Many systems of practical importance can only be described by complex models. For example, vibrations and sound waves exhibit both time and space dependence and are hence modelled by partial differential equations. This means that the state variables that evolve on infinite-dimensional spaces. (This is contrast to systems with ordinary equation models where the state is in Rn.) Systems modelled by delay-differential equations also evolve on an infinite-dimensional space. The dynamics may be complicated by the use of smart materials, such as piezo-electrics and shape memory alloys which have hysteretic and highly nonlinear behaviour.
An estimation of the state is often needed despite imperfect information. Estimators are mathematical algorithms that use data to produce a prediction of the state of a system. Typically both a mathematical model and measured data are available. Both contain errors. Examples include lake temperatures and currents, bridge vibrations, electrical networks and battery charge. The fundamental issue in estimation is to use data to yield the best estimate in the presence of uncertainty.
There are a number of projects available for qualified graduate students, in the following areas:
1. Systems with constraints. Many physical systems have dynamics that depend on space and time, coupled to equilibrium conditions. Examples include lithium ion cells, electric circuits, mechatronic systems with coupled mechanical/electromagnetic fields. Checkable conditions for existence of a solution and well-posedness are needed. Estimator and controller design have further issues.
2. Estimation of nonlinear systems. Existing design of estimators for nonlinear systems are largely heuristic. Popular methods sometimes perform well, but in other situations the estimator diverges. The use of formal cost functions in design of an optimal estimator is a challenging issue. One tool is machine learning to achieve real-time estimators for complex systems.
3. Optimal sensor design. Sensor placement and sensor shape design improves estimator accuracy without additional hardware. An integrated theory for sensor/estimator design for nonlinear systems requires research in several directions.
4. Control of nonlinear partial differential equations. Obtaining stabilizing controllers for systems with nonlinear partial differential equation models has many challenging problems.
5. Scientific computation. Using typical partial differential equation approximations in controller design suffers from the deficiency that the control signal (and disturbances) can affect higher modes which can lead to instability if the effect of the higher modes is not handled correctly. Although a wide body of results exist for linear parabolic systems, these issues are largely unaddressed for lightly damped waves and nonlinear systems. A closely related field is reduced-order modelling.
All accepted students will be fully funded. For more details, contact me.