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Second-Order Ordinary Differential Equations
Published in Brian Vick, Applied Engineering Mathematics, 2020
The linearized stability technique developed for one-dimensional systems in Section 10.1 is extended to two-dimensional systems. An autonomous system is one with no external driving forces and thus, no explicit time dependence. The general form for a second-order system isdxdt=f(x,y)dydt=g(x,y)
Critical Points, Direction Fields and Trajectories
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
A little more terminology: When dealing with direction fields and trajectories for a standard 2×2 autonomous system, we refer to the plane on which we sketch the direction field and/or trajectories as the phase plane (as opposed to the “XY–plane” or “X1X2–plane” plane or …). If we sketch an ‘enlightening’ representative sample of trajectories on the phase plane, then this sketch is said to be a phase portrait of the system. At this point, we are using direction fields to sketch phase portraits, so we are getting phase portraits superimposed on direction fields. If a phase portrait does not have an accompanying direction field to indicate direction of travel along the trajectories, then you should have little arrows on the trajectories to indicate the direction of travel for each trajectory.
Empirical studies of methods for safety and security co-analysis of autonomous boat
Published in Stein Haugen, Anne Barros, Coen van Gulijk, Trond Kongsvik, Jan Erik Vinnem, Safety and Reliability – Safe Societies in a Changing World, 2018
Erik Nilsen Torkildson, Jingyue Li, Stig Ole Johnsen, Jon Arne Glomsrud
The security and safety co-analysis methods can generally be classified into three categories (Kriaa et al., 2015). One category is generic approach, such as FMVEA (Schmittner et al., 2014a) and Fault Tree Analysis (Kornecki and Liu, 2013). Another category is model-based graphical methods, such as CHASSIS (Raspotnig et al., 2012) and method using Bayesian Belief Networks (Kornecki et al., 2013). The third category is model-based non-graphic methods, such as STPA (Young and Leveson, 2013) and unified framework (Asare et al., 2013). Autonomous systems are often cyber-physical systems that integrate computation, networking, and physical processes. In addition, autonomous systems need to have proper situation awareness using various sensors, and need to make correct decisions based on the sensor information. Thus, we decided to evaluate one method that is relevant to cyber-physical system in each category mentioned in (Kriaa et al., 2015). We chose FMVEA, CHASSIS, and STPA plus STPA-Sec, because FMVEA and CHASSIS are shown to be applicable to automotive cyberphysical systems (Schmittner et al., 2015), and STPA plus STPA-Sec focuses strongly on software dependent systems.
Adaptive sliding mode based fault tolerant control of wheeled mobile robots
Published in Automatika, 2023
Autonomous systems, including mobile robots, are widely used for many different tasks, such as search and rescue, monitoring, human–robot interaction tasks, etc. As the usage of these systems increases, the expected performance of these systems also increases. There are two major categories to consider for mobile robots: non-holonomic and holonomic. Wheeled Mobile Robots (WMR) are electromechanical systems that use activation torques to drive their wheels and are classified as non-holonomic. Autonomous systems, including mobile robots, are vulnerable to faults and external disturbances. Actuator, sensor, and controller (system) faults are the three major categories into which these faults can be divided. These faults can be amplified during the process of the system. It is crucial that the robot be able to behave tolerantly in the event of actuation and/or sensor faults. Therefore, Fault-Tolerant Control (FTC) systems are widely used today in a variety of fields, including the automotive and electronic industries, unmanned vehicle control, and even space research [1].
The spectral characterisation of reduced order models in chemical kinetic systems
Published in Combustion Theory and Modelling, 2022
Mauro Valorani, Riccardo Malpica Galassi, Pietro Paolo Ciottoli, Habib Najm, Samuel Paolucci
Let us assume that the multi-scale dynamics of a finite-dimensional system, whose state is defined by the vector , is governed by a set of ODEs of the form given the vector field , which can be linear/non-linear with respect to the state vector . We limit our discussion to autonomous systems, i.e. systems defined by a vector field that does not depend explicitly on time.2 In both circumstances, a time-scale characterisation of the dynamics can be carried out in terms of eigenvalues and right/left eigenvectors of the Jacobian matrix of .