Research Interests
My research interests focus on the general areas of nonlinear dynamical systems and control theory. Some of the topics that we are currently investigating along with my students include:
Analysis and Design of Nonlinear Sampled-Data Control Systems
Most systems of interest to control engineers work in continuous-time. Modern controllers, however, typically use modern digital technology for their implementation. A sample and hold device is used to provide the interface between the continuous-time plant and the the digital controller. For linear plants, sampled-data controller design is well understood, but nonlinear plants continue to be a challenge. There are two different approaches to design a sampled-data controller for a nonlinear plant: the first one consists of approximating a continuous-time controller at a sufficiently fast sampling rate. Hardware limitations, however, often impose restrictions on the sampling rate, making this approach ineffective. The second approach consists of discretizing the plant model and then proceed to design a discrete-time control law. The difficulty in this case is that finding the discrete-time model of a continuous-time plant requires the explicit analytical solution of a nonlinear differential equation that is often non-existent in analytical form. Virtually all the published work on the subject consider single-rate controllers, i.e. the sampling rate and the input measurement channels are equal. In many cases, however, the hardware restrictions of the input and measurement sampling rates are essentially different. We are investigating the ``multirate'' sampled-data control design problem, where we assume ``slow'' measurement rates and fast input samples. We have shown that multirate controllers are capable of stabilizing a nonlinear plant approximating the performance of a continuous time controller using a slower rates when compared to single rate controllers. The goal of our research is to develop a methodology for multirate sampled-data nonlinear control design.Nonlinear Observer Design
For a variety of reasons, it is often important to reconstruct the ``state'' of a dynamical system. The state is an internal variable that collects important information about the evolution of a dynamical system. Most techniques used in control design as well as virtually all problems of systems analysis assume that the state is available in real time. Unfortunately, the state is usually too expensive or impossible to measure and some form of estimation or reconstruction is necessary. The device used in the reconstruction is known as an ``observer''. The idea is to use the mathematical model of the system to reconstruct the state from input-output measurement. For linear time-invariant systems this problem has a well established solution. Despite recent advances, nonlinear systems on the other hand, represent a difficult challenge and nonlinear state reconstruction is a problem that remains largely unsolved.We recently proposed a new way to look at the observer problem and showed that reconstructing the state of a dynamical system is equivalent to stabilizing a feedback system of a certain form. Our early work was focused on estudying the robust estimation problem, understood as reconstructing the state of a plant in the presence of measurement noise and model uncertainty. More recently, We have extended our work to the problem of designing observers for nonlinear Lipschitz systems, a long standing unsolved problem in control theory, and observer design for sampled-data systems. The goal of this research is to obtain a technique for robust observer design for large classes of nonlinear systems.
Control and Synchronization of Chaotic Dynamical Systems
We are interested in the following problem: consider two chaotic dynamical systems, referred to as the master and slave systems. Under what conditions is it possible to make the trajectories of the slave system converge to those of the master system? Moreover, Suppose that the slave system is not exactly equal to the master; under these conditions: is it still possible to for the two subsystems to have identical trajectories, even starting from different initial conditions?
The problem above described is know as chaotic synchronization. Chaotic synchronization has a number of important theoretical implications, and it was proposed in 1990 as a was of ensuring secure communications. We have proposed a very simple synchronization method which views the synchronization problem as analogous to an observer design problem and thus exploited our experience on the observer design problem. We are currently investigating the effects of measurement delay as well as further work on robust synchronization.
Biomedical applications of contrrol
There is growing interest in genetic-based equipment for clinical diagnosis, forensic testing, population screening, etc. DNA analysis, in particular, can be used in a multiplicity of important applications such as cancer diagnosis, virus detection, screening for disease predisposition, etc. Typically, when processing a DNA sample, there is a need to increase the amount of DNA to accomodate for instrument limits of detection. A popular technique for genetic amplification is called ``Polymerase Chain Reaction'' (or PCR), which produces an exponential replication of the DNA sample. The PCR process requires that a mixture of DNA and other reagents be subjected to precise temperature cycling conditions consisting of the following steps:
Control of Large Industrial Facilities
Many industrial plants require the use of a cogeneration station to produce both steam and electric power. A typical plant utilizes a complex header system for steam distribution that receives steam from boiler system that might include utility-type (UB) boilers, CO-type boilers and once-through steam generators (OTSG). The steam is then distributed through the header system to several steam turbines to generate electricity. The overall plant can be viewed from a control point of view as a rather complex, nonlinear, interconnected system. In an on-going collaboration with industrial partner Syncrude Canada LTd. we are investigation advanced control technique for cogeneration systems. specific problem that we are considering include: