Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

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

Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network

A control scheme called repetitive control is proposed, in which the controlled variables follow periodic reference commands. A high-accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period L into the closed-loop system. Sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small-gain theorem and the stability theorem for time-lag systems. Synthesis algorithms are presented by both the state-space approach and the factorization approach. In the former approach, the technique of the Kalman filter and perfect regulation is utilized, while coprime factorization over the matrix ring of proper stable rational functions and the solution of the Hankel norm approximation are used in the latter one. >

/pdf/repetitive-control-system-a-new-type-servo-system-for-56qvga9fg5.pdf

Repetitive control system: a new type servo system for periodic exogenous signals

This paper presents a new framework for hybrid sampled data control systems. Instead of considering the state only at sampling instants, this paper introduces a function piece during the sampling period as the state and gives an infinite-dimensional model with such a state space. This gives the advantage that sampled data systems with built-in intersample behavior can be regarded as linear, time-invariant, discrete-time systems. As a result, the approach makes it possible to introduce such time-invariant concepts as transfer functions, poles, and zeros to the sampled data systems even with the presence of the intersample behavior. In particular, tracking problems can be studied in this setting in a simple and unified way, and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions. This leads to the internal model principle for sampled data systems. >

/pdf/a-function-space-approach-to-sampled-data-control-systems-4vr3u3a09o.pdf

A function space approach to sampled data control systems and tracking problems

This paper introduces the concept of frequency response for sampled-data systems and explores some basic properties as well as its computational procedures. It is shown that 1) by making use of the lifting technique, the notion of frequency response can be naturally introduced to sampled-data systems in spite of their time-varying characteristics, 2) it represents a frequency domain steady-state behaviour, and 3) it is also closely related to the original transfer function representation via an integral formula. It is shown that the computation of the frequency response can be reduced to a finite-dimensional eigenvalue problem, and some examples are presented to illustrate the results.

/pdf/frequency-response-of-sampled-data-systems-dyd7kqwwdy.pdf

Frequency response of sampled-data systems

The author presents a novel framework for hybrid sampled-data control systems. Instead of considering the state only at sampling instants, the author introduces a function piece during the sampling period as the state and gives an infinite-dimensional model with such a state space. This provides the advantage that sampled-data systems with built-in intersample behavior can be regarded as linear, time-invariant, discrete-time systems. As a result, the approach makes it possible to introduce such time-invariant concepts as transfer functions, poles, and zeros even in the presence of the intersample behavior. In particular, tracking problems can be studied in this setting in a simple and unified way, and stationary ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions. >

New approach to sampled-data control systems-a function space method

This paper studies the notion of frequency response for sampled-data systems, and explores some basic properties as well as its computational procedures. It is shown that: 1) by the lifting technique the notion of frequency response can be naturally justified for sampled-data systems in spite of their time-varying characteristics; 2) it represents a frequency domain steady state behavior; and 3) it is also closely related to the original transfer function representation via an integral formula. It is shown that the computation of the frequency response can be reduced to a finite-dimensional eigenvalue problem, and some examples are presented to illustrate the results. >

/pdf/frequency-response-of-sampled-data-systems-1g9mxxm9qi.pdf

Yutaka Yamamoto

Papers

Repetitive control system: a new type servo system for periodic exogenous signals

A function space approach to sampled data control systems and tracking problems

Frequency response of sampled-data systems

New approach to sampled-data control systems-a function space method

Frequency response of sampled-data systems