LEMS and Electrical Sciences Seminar
Abstract: The needs of our society are imposing ever increasing demands for efficient intelligent methodologies in control and automation, signal processing and communication networks. It is important to learn how to build systems that exhibit high degrees of autonomous behavior. To rise to the challenge, significant progress in the area of hybrid systems is needed. Hybrid systems arise from the interaction of discrete planning algorithms and continuous processes, and as such, they provide the basic framework and methodology for the analysis and synthesis of autonomous and intelligent systems.
Hybrid systems contain two distinct types of components, subsystems with continuous dynamics and subsystems with discrete event dynamics, that interact with each other. Such systems are important in a variety of contexts: Hybrid systems frequently arise from computer aided control of continuous processes in manufacturing, communication networks, autopilot design, computer synchronization, traffic control, and industrial process control, for example. Another important way in which hybrid systems arise is from the hierarchical organization of complex control systems. In these systems, a hierarchical organization helps manage complexity and higher levels in the hierarchy require less detailed models (discrete abstractions) of the functioning of the lower levels, necessitating the interaction of discrete and continuous components. Examples of such systems include flexible manufacturing and chemical process control systems, interconnected power systems, intelligent vehicle highway systems, air traffic management systems, computer communication networks.
The study of hybrid control systems is essential in designing sequential supervisory controllers for continuous systems, and it is central in designing intelligent control systems with a high degree of autonomy. The investigation of hybrid systems is creating a new and fascinating discipline bridging control engineering, mathematics and computer science.
In this talk, after an introduction to intelligent and hybrid systems, analysis and synthesis methodologies for hybrid control systems will be discussed. Discrete event systems will also be further discussed using Petri nets and a computationally efficient approach for the design of supervisory controllers, that uses the place invariants of the net will be presented.
Stochastic Systems Seminar
Abstract: The limit of the principal eigenvalue $\lambda^{\varepsilon}$ and the associated normalized eigenvector $\phi^{\varepsilon} $ for the operator of form $$ G^{\varepsilon} f = \varepsilon L f + \frac{1}{\varepsilon} V f $$ is in general analyzed through the transformation $ \mu^{\varepsilon} = \varepsilon \lambda^{\varepsilon}, W^{\varepsilon} = \varepsilon \log \phi^{\varepsilon} $. It is known that the limiting equation has multiple solutions in general. We describe solutions of the limiting equation for a particular case. We also show some examples that the limit of $ W^{\varepsilon}, \mu^{\varepsilon}$ exists, however, there is lack of uniqueness of the solutions for the limiting equation.
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: Numerical Simulations often provide strong evidences for the existence and stability of discrete shocks for certain finite difference schemes approximating conservation laws. In this talk, we present a framework for converting such numerical observations to mathematical proofs. The framework is applicable to conservative schemes approximating stationary shocks of one dimensional scalar conservation laws. The numerical flux function of the scheme is assumed to be twice differentiable but the scheme can be nonlinear and of any order of accuracy. To prove existence and stability, we show that it would suffice to verify some simple inequalities, which can usually be done using computers. As examples, we use the framework to give an unified proof of the existence of continuous discrete shock profiles for a modified first order Lax-Friedrichs scheme and the second order Lax-Wendroff scheme. We also show the existence and stability of discrete shocks for a third order weighted ENO scheme.
Department of Mathematics Colloquium
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