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

Controlling complex networks is of paramount importance in science and engineering. Despite the recent development of structural controllability theory, we continue to lack a framework to control undirected complex networks, especially given link weights. Here we introduce an exact controllability paradigm based on the maximum multiplicity to identify the minimum set of driver nodes required to achieve full control of networks with arbitrary structures and link-weight distributions. The framework reproduces the structural controllability of directed networks characterized by structural matrices. We explore the controllability of a large number of real and model networks, finding that dense networks with identical weights are difficult to be controlled. An efficient and accurate tool is offered to assess the controllability of large sparse and dense networks. The exact controllability framework enables a comprehensive understanding of the impact of network properties on controllability, a fundamental problem towards our ultimate control of complex systems.

/pdf/exact-controllability-of-complex-networks-3k2ioruanh.pdf

Exact controllability of complex networks

SUMMARY
This paper considers the containment control problems for both continuous-time and discrete-time multi-agent systems with general linear dynamics under directed communication topologies. Distributed dynamic containment controllers based on the relative outputs of neighboring agents are constructed for both continuous-time and discrete-time cases, under which the states of the followers will asymptotically converge to the convex hull formed by those of the leaders if, for each follower, there exists at least one leader that has a directed path to that follower. Sufficient conditions on the existence of these dynamic controllers are given. Static containment controllers relying on the relative states of neighboring agents are also discussed as special cases. Copyright © 2011 John Wiley & Sons, Ltd.

Distributed containment control of multi‐agent systems with general linear dynamics in the presence of multiple leaders

Combined with backstepping techniques, an observer-based adaptive consensus tracking control strategy is developed for a class of high-order nonlinear multiagent systems, of which each follower agent is modeled in a semi-strict-feedback form By constructing the neural network-based state observer for each follower, the proposed consensus control method solves the unmeasurable state problem of high-order nonlinear multiagent systems The control algorithm can guarantee that all signals of the multiagent system are semi-globally uniformly ultimately bounded and all outputs can synchronously track a reference signal to a desired accuracy A simulation example is carried out to further demonstrate the effectiveness of the proposed consensus control method

Observer-Based Adaptive Backstepping Consensus Tracking Control for High-Order Nonlinear Semi-Strict-Feedback Multiagent Systems

In this paper, the cooperative adaptive fault tolerant fuzzy tracking control (CAFTFTC) problem of networked high-order multiagent with time-varying actuator faults is studied, and a novel CAFTFTC scheme is proposed to guarantee that all follower nodes asymptotically synchronize a leader node with tracking errors converging to a small adjustable neighborhood of the origin in spite of actuator faults. The leader node is modeled as a higher order nonautonomous nonlinear system. It acts as a command generator giving commands only to a small portion of the networked group. Each follower is assumed to have nonidentical unknown nonlinear dynamics, and the communication network is also assumed to be a weighted directed graph with a fixed topology. A distributed robust adaptive fuzzy controller is designed for each follower node such that the tracking errors are cooperative uniform ultimate boundedness (CUUB). Moreover, these controllers are distributed in the sense that the controller designed for each follower node only requires relative state information between itself and its neighbors. The adaptive compensation term of the optimal approximation errors and external disturbances is adopted to reduce the effects of the errors and disturbances, which removes the assumption that the upper bounds of unknown function approximation errors and disturbances should be known. Analysis of stability and parameter convergence of the proposed algorithm are conducted that are based on algebraic graph theory and Lyapunov theory. Comparing with results in the literature, the CAFTFTC scheme can minimize the time delay between fault occurrence and accommodation and reduce its adverse effect on system performance. In addition, the FTC scheme requires no additional fault isolation model, which is necessary in the traditional active FTC scheme. Finally, an example is provided to validate the theoretical results.

Cooperative Adaptive Fuzzy Tracking Control for Networked Unknown Nonlinear Multiagent Systems With Time-Varying Actuator Faults

Consensus problems for high-order continuous-time linear time-invariant swarm systems with directed interaction topologies are investigated. A new method to deal with consensus problems is proposed based on state space decomposition. Two subspaces of a complex space are introduced, namely a consensus subspace and a complement consensus subspace. Based on this decomposition, necessary and sufficient conditions for consensus and consensualizability are presented respectively. An approach for determining and designing the consensus function which may be time-varying is shown. The applications of theoretical results in multi-agent supporting systems are studied.

Consensus problems for high-order linear time-invariant swarm systems

In this study the swarm stability problem of high-order linear time-invariant (LTI) swarm systems with directed graph topology is dealt with. Consensus can be regarded as a specific type of swarm stability problem. Necessary and sufficient conditions for both swarm stability and consensus are presented. These conditions depend on the graph topology, the dynamics of agents and the interactions among the neighbours. Simulation instances are shown to illustrate the theoretical results.

Brief paper swarm stability of high-order linear time-invariant swarm systems

This paper discusses the controllability problem of complex networks. It is shown that almost any weighted complex network with noise on the strength of communication links is controllable in the sense of Kalman controllability. The concept of almost controllability is elaborated by both theoretical discussions and experimental verifications.

On Almost Controllability of Dynamical Complex Networks with Noises

The controllability problem of high-order linear time-invariant (LTI) continuous-time swarm systems is investigated. A necessary and sufficient condition for complete controllability of homogeneous swarm systems is given. The controllability of graph is important and is analysed in detail. The concept of formation controllability is proposed and the relationship between complete controllability and formation controllability is studied.

Formation controllability of high-order linear time-invariant swarm systems

This paper presents a novel approach for clustering, which is based on quasi-consensus of dynamical linear high-order multiagent systems. The graph topology is associated with a selected multiagent system, with each agent corresponding to one vertex. In order to reveal the cluster structure, the agents belonging to a similar cluster are expected to aggregate together. To establish the theoretical foundation, a necessary and sufficient condition is given to check the achievement of group consensus. Two numerical instances are furnished to illustrate the results of our approach.

/pdf/a-novel-clustering-method-based-on-quasi-consensus-motions-1kh6mswy08.pdf

Ning Cai

Papers

Consensus problems for high-order linear time-invariant swarm systems

Brief paper swarm stability of high-order linear time-invariant swarm systems

On Almost Controllability of Dynamical Complex Networks with Noises

Formation controllability of high-order linear time-invariant swarm systems

A Novel Clustering Method Based on Quasi-Consensus Motions of Dynamical Multiagent Systems