My interests are in the area of nonlinear control theory, with emphasis on stability theory and robust and optimal control. Robust control is the branch of control theory that takes explicit account of the fact that the mathematical models used in control design are never perfect. This is widely recognized as a very important problem. Indeed, due to the existence of model uncertainties, controllers which operate very well in computer simulations often fail to provide adequate performance when put to work with the actual plant. For linear time-invariant systems, robust control has reached certain degree of maturity. My long term interest is in the formulation of a comprehensive theory of robust control for nonlinear systems, which can deal with complex industrial systems.
Over the past year my work focused on three problems: sampled-data control, nonlinear observers, and, chaos control and synchronization.
Sampled data nonlinear control is a relatively new area that takes account of the fact that exact discrete-time models of continuous-time plants cannot be obtained due to the lack of existence of closed form solutions of nonlinear differential equations. For the past three years we have been working on the development of a theory of multirate sampled-data control. We were able to show that multirate systems can significantly improve the performance of single rate controllers. We are currently working on the formulation of a design technique for multirate nonlinear controllers.
Work on nonlinear observers focused on the use of a new observer structure, which I recently introduced. In cooperation with my graduate students, we have developed a complete observer design procedure for Lipschitz systems along with extensions to discrete-time and multirate systems. Currently our work is focused on an extension of previous results to a larger class of systems known as "one-sided Lipschitz."
Nonlinear dynamical systems can have very interesting properties. One of those properties is called "chaos". Chaos is characterized by critical dependence on initial conditions. My work in this area focuses on how to synchronize the trajectories of two identical dynamical systems. If successful, chaotic systems can be used to send communication message through a secure channel.