MASALAH EIGEN DAN EIGENMODE MATRIKS ATAS ALJABAR MIN-PLUS

Eigen problems and eigenmode are important components related to square matrices. In max-plus algebra, a square matrix can be represented in the form of a graph called a communication graph. The communication graph can be strongly connected graph and a not strongly connected graph. The representation matrix of a strongly connected graph is called an irreducible matrix, while the representation matrix of a graph that is not strongly connected is called a reduced matrix. The purpose of this research is set the steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and also eigenmode of the regular reduced matrix over min-plus algebra. Min-plus algebra has an ispmorphic structure with max-plus algebra. Therefore, eigen problems and eigenmode matrices over min-plus algebra can be determined based on the theory of eigenvalues, eigenvectors and eigenmode matrices over max-plus algebra. The results of this research obtained steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and eigenmode algorithm of the regular reduced matrix over min-plus algebra

Distributed Observer Design for Linear Systems under Time-Varying Communication Delay

The paper investigates the state estimation problem of general continuous-time linear systems with the consideration of time-varying communication delay. A solution is proposed in terms of the networked distributed observer, which consists of multiple local observers. Each local observer relies on only part of the system output and exchanges information with neighbors through undirected links modeled by a prespecified communication graph. A simple approach for computing observer parameters is presented by solving a parametric algebraic Riccati equation. Furthermore, by the Lyapunov–Krasovskii stability theorem, an upper bound of the delay could be calculated explicitly and together with the conditions of joint observability and connectivity of the communication graph; the resulting distributed observers work coordinately to achieve an asymptotic estimate of the full plant state. An illustrative example is provided to confirm the analytical results.

Dynamic Routing and Coordination of Cluster for Unmanned Aerial Vehicle (UAV) Swarms

The flying networks provide an efficient solution for a wide range of military and commercial purposes. The demand for portable and flexible communication is directed towards a quick growth in interaction among unmanned aerial vehicles (UAVs). Due to the frequent change in topology and high mobility of vehicles, routing and coordination becomes a challenging task. To maximize the throughput of the network, this study addresses the UAV swarm’s problems related to the coordination and routing and defines the proposed solution to solve these issues. For this, a network is assumed which contains an equal number of dynamic vehicles. It also presents the communication graph concept of UAVs and designs a fixed-wing UAV model to improve the efficiency of the network in terms of throughput. Furthermore, the proposed algorithm based on Cauchy particle swarm optimization (CPSO) aims towards the better performance of UAV swarms and aims to solve the combinatorial problem. The simulation results show and confirm the performance of the proposed algorithm.

Observer-Based Consensus Control for Heterogeneous Multi-Agent Systems with Output Saturations

This paper studies the consensus problem for heterogeneous multi-agent systems with output saturations. We consider the agents to have different dynamics and assume that the agents are neutrally stable and that the communication graph is undirected. The goal of this paper is to achieve the consensus for leaderless and leader-following cases. To solve this problem, we propose the observer-based distributed consensus algorithms, which consists of three parts: the nonlinear observer, the reference generator, and the regulator. Then, we analyze the consensus based on the Lasalle’s Invariance Principle and the input-to-state stability. Finally, we provide numerical examples to demonstrate the validity of the proposed algorithms.

On Constructing t -Spanner in IoT under SINR

Following the recent advances in the Internet of Things (IoT), it is drawing lots of attention to design distributed algorithms for various network optimization problems under the SINR (Signal-to-Interference-and-Noise-Ratio) interference model, such as spanner construction. Since a spanner can maintain a linear number of links while still preserving efficient routes for any pair of nodes in wireless networks, it is important to design distributed algorithms for spanners. Given a constant t > 1 as the required stretch factor, the problem of our concern is to design an efficient distributed algorithm to construct a t -spanner of the communication graph under SINR such that the delay for the task completion is minimized, where the delay is the time interval between the time slot that the first node commences its operation to the time slot that all the nodes finish their task of constructing the t -spanner. Our main contributions include four aspects. First, we propose a proximity range and proximity independent set (PISet) to increase the number of nodes transmitting successfully at the same time in order to reduce the delay. Second, we develop a distributed randomized algorithm SINR-Spanner to construct a required t -spanner with high probability. Third, the approximation ratio of SINR-Spanner is proven to be a constant. Finally, extensive simulations are carried out to verify the effectiveness and efficiency of our proposed algorithm.

Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

In a previous paper we defined the black and white SAT problem which has exactly two solutions, where each variable is either true or false. We showed that black and white 2-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonboglár model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a black and white SAT problem. We prove a powerful theorem, the so called transitions theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as black and white SAT problems. We show that the Balatonboglár model is between the strong and the weak model, and it generates 3-SAT problems, so the Balatonboglár model represents strongly connected communication graphs as black and white 3-SAT problems. Our motivation to study these models is the following: The strong model generates a 2-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonboglár model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonboglár model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: the SAT problem and directed graphs.

Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.