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 are complicated by the use of smart materials, such as piezo-electrics and shape memory alloys, which are of interest to control structures and acoustic systems. Their behaviour is hysteretic and highly nonlinear. Analysis and simulation of these systems is often not straightforward; the research in my group is focused on the even more challenging problem of designing controllers to alter the dynamics so as to achieve such goals as improved settling time and tracking of a reference signal.

Some controller design theory for infinite-dimensional systems is analogous to that for finite-dimensional systems, but the controller is now an operator that needs to be approximated to obtain an implementable finite-dimensional control law. The approximations that are used in simulation and controller design suffer from the deficiency that the control signal (and also disturbances) can affect higher modes which can lead to instability if the effect of the higher modes is not handled correctly.

The zeros of a control system have the same importance in controller design as the eigenvalues do for stability analysis. Despite increasingly sophisticated theories for infinite-dimensional systems, little attention has been paid to the zeros or minimum-phase properties. This was a serious deficiency, particularly since the zeros of approximations are often quite different from the original model, even when the eigenvalues are close. Recent collaborative work has been developing a theory for zeros of infinite-dimensional systems. An open problem, related to the work discussed above, is determining which approximations reproduce zeros correctly, and when the zero accuracy predicts good controller design.

Current projects include the following:

1. Optimal actuator/sensor location: For systems with time and space dependence, the location of the controller hardware can be selected. The choice of locations affects the performance and also the efficiency of the final controlled system. In many cases, the hardware can not be easily moved once installed. For example, piezo-electric patches are very diffiicult to remove once bonded to a structure. This motivates a mechatronic approach where the locations are included as part of the controller design process. We have been working on obtaining theoretical results for when such problems are well-posed, and also when approximations can be used to compute the controller and locations. Recent work has also been concerned with the development of algorithms suitable for large-scale approximating models.

2. Controller design for nonlinear partial differential equations: Obtaining stabilizing controllers for systems with nonlinear partial differential equation models is a new area with many interesting problems. Few conditions even exist for when the stability of the linearized system predicts stability of the nonlinear system.

There are projects available for qualified graduate students. All accepted students will be fully funded. For more details, contact me.