PhD Topics

(I am looking for students to work on some of these topics)

Note: If you are interested in a topic that is not listed above but which falls in the area of nonlinear control theory, I would be very happy to discuss that with you.
Networked Control Systems:

In networked control systems (NCS), one or more dynamical systems are controlled by feedback over a communication network. The transmission capacity of the communication network is limited. This limits the number of bits or packets per second which can be transported via the network and, consequently, restricts the achievable performance. This area has grown rapidly in the last few years with the emergence of applications ranging from micro-electromechanical chips and Internet congestion protocols to ``drive-by-wire" systems. Many attractive advantages of introducing a communication network, like high system testability and resource utilization, as well as low weight, space, power and wiring requirements, motivate the research on the NCS. On the other hand, networks can introduce unreliable and time-dependent levels of service in terms of delays, jitter, data packet losses, and so on.

This project will deal with the development of methods for analysis and controller design of networked control systems. In particular, we intend to further develop a framework for NCS design proposed by myself and A.R. Teel.

Extremum seeking control:

In many control applications the reference-to-output map has an extremum and the control objective is to regulate the output close to this extremum. The main objective in extremum seeking control is to force the solutions of the closed loop system to eventually converge to a set in the state space where its output achieves an extremum and to do so without precise knowledge about the reference-to-output map. While this is an old topic, local stability of a class of extremum seeking controllers was proved for the first time by Krstic and Wang in 2000. We have extended these results and proved semiglobal-practical stability of the same class of extremum seeking controllers under natural assumptions.

This project will concentrate on providing guidelines of extremum seeking control design and tuning, as well as stability and robustness analysis of such systems. Furthermore, applications of extremum seeking control to particular engineering problems will be explored.

Analysis and design of reset systems:

Reset controllers are an example of nonlinear controllers that allow for extra design flexibility and they are motivated by the so called Clegg integrator introduced in 1958. This device is a particular type of a nonlinear integrator that operates in the same manner as the linear integrator whenever its input and output have the same sign and it resets its output to zero otherwise. Its describing function has the same magnitude plot as the linear integrator but it has a phase lag of only 38.1 degrees compared to the lag of 90 degrees for a linear integrator. This feature can be used to provide more flexibility in controller design. Indeed, it was recently shown that reset controllers may help remove some of the fundamental performance limitations of linear controllers. While these examples motivate the consideration of reset controllers, there are still no systematic methods for design of such systems.

This project aims at developing reset controller design methods that remove some of the fundamental performance limitations of linear controllers.

Digital nonlinear controller design with applications:

Due to prevalence of computer-controlled systems, it is of utmost importance to be able to design digital controllers for nonlinear continuous-time plants. The main obstacle to the controller design is that the exact discrete-time or sampled-data model of the system is typically unknown even when the underlying continuous-time model is known. Recently, I have started developing a framework for digital nonlinear controller design based on approximate discrete-time plant models. This framework uses ideas from numerical integration literature to develop digital controllers for nonlinear continuous-time plants. We are able to show in most examples that our method outperforms some other available methods for digital controller design. However, a range of open questions remain unsolved in this area.

Asymptotic methods for nonlinear systems with disturbances:

There is a range of tools (such as averaging, singular perturbations, slowly varying systems methods, etc) in the area of dynamical systems that have found use in nonlinear control theory (such as in vibrational control, adaptive control, control of systems with high frequency unmodelled dynamics, gain scheduling, etc.). Most of the available tools are given for systems without disturbances although systems with disturbances are prevalent in control theory. I have recently started generalizing a range of dynamical systems tools to investigate the so called input-to-state stability (ISS) of systems with disturbances, which is a particular definition of bounded-input bounded-output stability. We have obtained a rather general and unifying theory for investigation of the ISS property. I am interested in further generalizing these tools to investigation of properties other than ISS. Moreover, I would like to explore how these tools can be used to improve controller design techniques for systems with disturbances using vibrational control, adaptive control, control of systems with high frequency unmodelled dynamics, gain scheduling, etc.

Digital nonlinear observer design with applications:

Very often one does not measure all the state variables that are needed for a good controller operation but rather estimates them from a few measurements. The device that estimates the state of the system from its outputs (measurements) is called an observer. It is usually the case that the plant is a continuous-time system whereas the observer is a digital system (usually written as a part of the control algorithm programmed on a computer). However, the available theory for observer design either assumes that the observer is a continuous-time (and not digital) system or that the exact discrete-time plant model is known. Both of these assumptions are typically not true if the plant model is nonlinear. I have recently started developing theory for controller design based on approximate discrete-time plant models. This project aims at developing a similar (actually dual) framework for digital observer design based on approximate discrete-time plant models.

Applications of nonlinear control theory:

Besides the theoretical developments in mathematical control theory that I am interested in, I would also like to explore how some of the recently developed nonlinear control techniques can be applied to control different systems. Some applications that I am interested in are listed below: