关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Mathematica--A System for Doing Mathematics by Computer.

This article introduces learning analytics dashboards that visualize learning traces for learners and teachers. We present a conceptual framework that helps to analyze learning analytics applications for these kinds of users. We then present our own work in this area and compare with 15 related dashboard applications for learning. Most evaluations evaluate only part of our conceptual framework and do not assess whether dashboards contribute to behavior change or new understanding, probably also because such assessment requires longitudinal studies.

/pdf/learning-analytics-dashboard-applications-32tue7p884.pdf

Learning analytics dashboard applications

A robust multivariable controller is introduced for a class of distributed parameter systems. The system to be controlled is given as \dot{x} = Ax + Bu, y = Cx in a Banach space. The purpose of the control, which is based on the measurement y , is to stabilize and regulate the system so that y(t) \rightarrow y_{r}, as t \rightarrow \infty , where y r is a constant reference vector. Under the assumptions that operator A generates a holomorphic stable semigroup, B is linear and bounded, C is linear and A -bounded, and the input and output spaces are of the same dimension; a necessary and sufficient condition is found for the existence of a robust multivariable controller. This controller appears to be a multivariable PI-controller. Also, a simple necessary criterion for the existence of a decentralized controller is derived. The tuning of the controller is discussed and it is shown that the I-part of the controller can be tuned on the basis of step responses, without exact knowledge of the system's parameters. The presented theory is then used as an example to control the temperature profile of a bar, with the Dirichlet boundary conditions.

Robust multivariable PI-controller for infinite dimensional systems

In this paper we consider robust output regulation of distributed parameter systems and the internal model principle. The main purpose is to generalize the internal model principle by Francis and Wonham for infinite-dimensional systems and clarify the relationships between different generalizations of the internal model. We also construct a signal generator capable of generating infinite-dimensional polynomially increasing signals.

Internal Model Theory for Distributed Parameter Systems

In this paper a robust regulation problem for infinite-dimensional systems with infinite-dimensional exosystems is discussed. The input and output spaces are also allowed to be infinite-dimensional. A new definition of internal model in terms of the controller parameters is given. It is shown that there exists a feedback controller containing an internal model of the exosystem, which robustly regulates the class of signals generated by the exosystem and strongly or weakly stabilizes the closed-loop system. As far as the authors know, the results are new even for finite-dimensional systems with infinite-dimensional exosystems.

Robust Regulation of Distributed Parameter Systems with Infinite-Dimensional Exosystems

In this paper the theory of robust output regulation of distributed parameter systems with infinite-dimensional exosystems is extended for plants with unbounded control and observation. As the main result, we present the internal model principle for linear infinite-dimensional systems with unbounded input and output operators. We do this for two different definitions of an internal model found in the literature, namely, the p-copy internal model and the $\mathcal{G}$-conditions. We also introduce a new way of defining an internal model for infinite-dimensional systems. The theoretic results are illustrated with an example where we consider robust output tracking for a one-dimensional heat equation with boundary control and pointwise measurements.

The Internal Model Principle for Systems with Unbounded Control and Observation

Solutions of social media enhanced learning are widely studied in Hypermedia Laboratory at Tampere University of Technology (TUT). In recent years Web 2.0 based social media services (e.g., Facebook®, LinkedIn®, Last.fm®, etc.) have become popular, especially among young people. Based on this phenomenon TUT Hypermedia researchers have developed a social networking site for TUT freshmen aiming to provide convenient tools for interaction and study support. The first idea was to offer a free-of-charge social web site in the context of learning Basic Engineering Mathematics at TUT. This was thought to be an efficient tool to get new students studies off to a good start as mathematics courses play a significant role. However, the prediction failed, which caused us to study students’ motivations for social network site usage in the study context. This paper describes research conducted in 2009. Moreover, a description of subsequent measures accomplished (e.g., web site development and social network analysis) at TUT is included. https://doi.org/10.34105/j.kmel.2010.02.005

/pdf/students-motivations-for-social-media-enhanced-studying-and-reqcxy9rnk.pdf

Seppo Pohjolainen

Papers

Robust multivariable PI-controller for infinite dimensional systems

Internal Model Theory for Distributed Parameter Systems

Robust Regulation of Distributed Parameter Systems with Infinite-Dimensional Exosystems

The Internal Model Principle for Systems with Unbounded Control and Observation

Students’ Motivations for Social Media Enhanced Studying and Learning