Fractional Differential Equations

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

http://repositorio.uchile.cl/bitstream/handle/2250/127037/Lyapunov-functions-for-fractional-order-systems.pdf?sequence=1

Lyapunov functions for fractional order systems

Neuromorphic computing has come to refer to a variety of brain-inspired computers, devices, and models that contrast the pervasive von Neumann computer architecture This biologically inspired approach has created highly connected synthetic neurons and synapses that can be used to model neuroscience theories as well as solve challenging machine learning problems The promise of the technology is to create a brain-like ability to learn and adapt, but the technical challenges are significant, starting with an accurate neuroscience model of how the brain works, to finding materials and engineering breakthroughs to build devices to support these models, to creating a programming framework so the systems can learn, to creating applications with brain-like capabilities In this work, we provide a comprehensive survey of the research and motivations for neuromorphic computing over its history We begin with a 35-year review of the motivations and drivers of neuromorphic computing, then look at the major research areas of the field, which we define as neuro-inspired models, algorithms and learning approaches, hardware and devices, supporting systems, and finally applications We conclude with a broad discussion on the major research topics that need to be addressed in the coming years to see the promise of neuromorphic computing fulfilled The goals of this work are to provide an exhaustive review of the research conducted in neuromorphic computing since the inception of the term, and to motivate further work by illuminating gaps in the field where new research is needed

A Survey of Neuromorphic Computing and Neural Networks in Hardware.

A recent set of overused population-based methods have been published in recent years. Despite their popularity, most of them have uncertain, immature performance, partially done verifications, similar overused metaphors, similar immature exploration and exploitation components and operations, and an insecure tradeoff between exploration and exploitation trends in most of the new real-world cases. Therefore, all users need to extensively modify and adjust their operations based on main evolutionary methods to reach faster convergence, more stable balance, and high-quality results. To move the optimization community one step ahead toward more focus on performance rather than change of metaphor, a general-purpose population-based optimization technique called Hunger Games Search (HGS) is proposed in this research with a simple structure, special stability features and very competitive performance to realize the solutions of both constrained and unconstrained problems more effectively. The proposed HGS is designed according to the hunger-driven activities and behavioural choice of animals. This dynamic, fitness-wise search method follows a simple concept of “Hunger” as the most crucial homeostatic motivation and reason for behaviours, decisions, and actions in the life of all animals to make the process of optimization more understandable and consistent for new users and decision-makers. The Hunger Games Search incorporates the concept of hunger into the feature process; in other words, an adaptive weight based on the concept of hunger is designed and employed to simulate the effect of hunger on each search step. It follows the computationally logical rules (games) utilized by almost all animals and these rival activities and games are often adaptive evolutionary by securing higher chances of survival and food acquisition. This method's main feature is its dynamic nature, simple structure, and high performance in terms of convergence and acceptable quality of solutions, proving to be more efficient than the current optimization methods. The effectiveness of HGS was verified by comparing HGS with a comprehensive set of popular and advanced algorithms on 23 well-known optimization functions and the IEEE CEC 2014 benchmark test suite. Also, the HGS was applied to several engineering problems to demonstrate its applicability. The results validate the effectiveness of the proposed optimizer compared to popular essential optimizers, several advanced variants of the existing methods, and several CEC winners and powerful differential evolution (DE)-based methods abbreviated as LSHADE, SPS_L_SHADE_EIG, LSHADE_cnEpSi, SHADE, SADE, MPEDE, and JDE methods in handling many single-objective problems. We designed this open-source population-based method to be a standard tool for optimization in different areas of artificial intelligence and machine learning with several new exploratory and exploitative features, high performance, and high optimization capacity. The method is very flexible and scalable to be extended to fit more form of optimization cases in both structural aspects and application sides. This paper's source codes, supplementary files, Latex and office source files, sources of plots, a brief version and pseudocode, and an open-source software toolkit for solving optimization problems with Hunger Games Search and online web service for any question, feedback, suggestion, and idea on HGS algorithm will be available to the public at https://aliasgharheidari.com/HGS.html .

Hunger games search: Visions, conception, implementation, deep analysis, perspectives, and towards performance shifts

Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems

Reliable asynchronous sampled-data filtering of T–S fuzzy uncertain delayed neural networks with stochastic switched topologies

Nonfragile asynchronous control for uncertain chaotic Lurie network systems with Bernoulli stochastic process

This paper proposes a new non-fragile stochastic control method to investigate the robust sampled-data synchronization problem for uncertain chaotic Lurie systems (CLSs) with time-varying delays. The controller gain fluctuation and time-varying uncertain parameters are supposed to be random and satisfy certain Bernoulli distributed white noise sequences. Moreover, by choosing an appropriate Lyapunov-Krasovskii functional (LKF), which takes full advantage of the available information about the actual sampling pattern and the nonlinear condition, a novel synchronization criterion is developed for analyzing the corresponding synchronization error system. Furthermore, based on the most powerful free-matrix-based integral inequality (FMBII), the desired non-fragile sampled-data estimator controller is obtained in terms of the solution of linear matrix inequalities. Finally, three numerical simulation examples of Chua's circuit and neural network are provided to show the effectiveness and superiorities of the proposed theoretical results.

Shouming Zhong

Papers

Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems

Reliable asynchronous sampled-data filtering of T–S fuzzy uncertain delayed neural networks with stochastic switched topologies

Nonfragile asynchronous control for uncertain chaotic Lurie network systems with Bernoulli stochastic process

Non-fragile sampled-data robust synchronization of uncertain delayed chaotic Lurie systems with randomly occurring controller gain fluctuation.

Design of sliding mode controller for a class of fractional-order chaotic systems